Approximation of conformal mappings by circle patterns

Appro ximatio n of conform al mappings b y cir cle patterns Ulrik e B ¨ uc king ∗ Abstract A circle pattern is a conﬁguration of circles in the plane whose combi- natorics is given b y a planar graph G such that to eac h vertex of G there corresponds a circle. If tw o vertic es are connected by an edge in G then the corresponding circles intersect w ith an intersec tion angle in (0 , π ) and these interse ction p oints can b e associated to the d ual graph G ∗ . Tw o sequences of circle patterns are employ ed to approximate a giv en conformal map g and its ﬁrst deriv ative. F o r the domain of g w e u se em b edded circle patterns where all circles ha ve the same radius ε n > 0 for a sequence ε n → 0 and where the intersection angles are uniformly b ounded. The image circle patterns h av e the same combinatorics and inters ection angles and are determined from b oundary conditions (radii or angl es) acco rd ing to the v alues of g ′ ( | g ′ | or arg g ′ ). The error is of order 1 / √ − log ε n . F o r qu asicrystallic circle p atterns the con vergence result is strengthened t o C ∞ -conv ergence on compact sub sets and an error of order ε n . 1 In tro d uction Conformal mappings constitute an imp orta nt class in the ﬁeld of complex analy- sis. T he y may be characterized by the fact that inﬁnitesima l cir cles are mappe d to inﬁnitesimal cir cles. Suitable discrete analo gs are of actual interest in the area of discr ete diﬀerential g eometry a nd its applications, see [BSSZ08]. Bill Thurston ﬁrst intro duce d in his talk [Thu85] the idea to use ﬁnite cir cles, in particular circ le pa ckings, to deﬁne a discrete confor mal mapping. Remember that an emb e dd e d planar cir cle p acking is a co nﬁguration o f closed disks with disjoint interiors in the pla ne C . Connecting the centers o f touc hing disks b y straight lines yields the tangency gr aph . Le t C 1 and C 2 be t wo circle pa ckings whose tangency graphs ar e c o mbinatorially the same. Then there is a mapping g C : C 1 → C 2 which maps the centers of circles of C 1 to the co rresp o nding centers of circles of C 2 and is an aﬃne map o n ea ch triangular region cor re- sp onding to three m utua lly tangent cir cles. V arious connectio ns b etw een circle packings a nd classica l complex analy sis hav e alre a dy b ee n studied. A bea utiful int r o duction a nd sur wa y is present ed by Stephenso n in [Ste05]. In par ticular, several res ults concerning conv erg ence, i.e. quantitative approximation of co n- formal mappings by g C , ha ve b een obtaine d, see [RS87, CR92, HS96, HS98]. ∗ Pa r tially supp orted by the D FG Research Unit “Polyhe dral Surfaces” and by the DFG Researc h Cent er Ma theon “Mathematics for key techno l ogies” 1 2 Ulrike B ¨ ucking The cla ss o f cir cle p att ern s gener alizes circle packings as for each circle pack- ing there is an a sso ciated orthogo nal circle pattern. Simply add a circle for e ach triangular face which passes throug h the three touching p oints. T o deﬁne a cir- cle pa ttern we use a planar gra ph as combinatorial da ta. The circles co r resp ond to vertices and the edges sp ec ify which circles should in ter sect. The intersection angles are given using a labelling on the edg es. Thus an edge corresp onds to a kite of tw o in ter secting circles as in Figure 1 (rig ht) . Moreover, for in ter ior vertices the kites corres po nding to the incident edg es hav e disjoint interiors and their unio n is ho meomorphic to a closed disk. Similarly as for circle packings, there are results on existence, rig idity , and construction of sp ec ia l circle pat- tern. See for exa mple [Riv94, Sc h97, AB00, BS04, BH03], where some of the references use a genera lized notion of cir cle pa tterns. Given tw o circle patterns C 1 and C 2 with the same combinatorics a nd inter- section ang le s, deﬁne a ma pping g C : C 1 → C 2 similarly a s for circle packings. Namely , take g C to map the centers o f circles and the intersection p o ints of C 1 corres p o nding to vertices and faces of G to the corre s p o nding centers of circles and intersection p oints of C 2 and extend it to an a ﬃne map on each kite. F or a given confor mal ma p g we use an a nalytic a pproach and sp ecify suit- able bo unda ry v alues for the r adius or the ang le function a ccording to | g ′ | or to arg g ′ resp ectively in order to deﬁne the (approximating) mapping s g C . Gener- alizing ideas of Schramm’s conv er gence pro o f in [Sch97] w e obtain conv erge nce in C 1 on c o mpact s ets if we take for C ( n ) 1 a s equence of isoradia l circle patterns (i.e. all radii are equal) with decr easing radii ε n → 0 which a pproximate the domain of g . F urthermor e, we assume that the intersection a ngles a re uniformly bo unded awa y fro m 0 and π . Note in particular tha t the combinatorics o f the circle pattern C ( n ) 1 may be irregular or change within the seq uence. Thu s o ur conv erg ence r esults a pplies to a cons ide r ably bro a der class of circle patterns a s the known results o f Schramm [Sch97], Matthes [Mat05], or Lan and Dai [LD07] for o rthogona l cir cle patter ns with sq uare gr id combinatorics. The ma in idea of the pro of is to cons ider a “nonlinear discre te Laplace equation” for the ra dius function. This equatio n turns out to b e a (go o d) approximation of a known linear La place equation a nd can b e used in the ca se of isoradia l circle patterns to compar e discrete and smo o th solutions of the corres p o nding elliptic proble ms , that is the logarithm o f the radii of C ( n ) 2 and log | g ′ | = Re(lo g g ′ ). Also, we o btain an a priori estimation of the approximation error of orde r 1 / p log(1 /ε n ). If the circle patterns C ( n ) 1 additionally have only a uniformly b ounded num- ber of diﬀerent edge directions, then the corr esp onding kite pa tterns a re qua- sicrystallic rhom bic em bedding s a nd the circle patterns C ( n ) 1 are ca lled qua- sicrystal lic [BMS05]. F or such embeddings we genera liz e an asymptotic devel- opment given by K eny on in [K en02] of a discrete Green’s function. Also , using similar ideas as Duﬃn in [Duf53], we gene r alize theor ems o f discr ete p otential theory concerning the regula rity of solutions of a discrete La place equation. W e then use these results together with the C 1 -conv erg ence for isoradia l circle pa t- terns and prove C ∞ -conv erg ence on compact s ets for a class of qua sicrystallic circle patterns. In this ca se the approximation error is of order ε n (or ε 2 n for square grid or hex a gonal c o mbinatorics and regula r intersection angles). The pro of generalizes a metho d used b y He and Sc hr amm in [HS98]. The article is or ganized a s follows: First we introduce and r emind the ter- Circle p a tterns 3 minology and s ome results on circle patterns in Section 2, fo c us ing in par ticular on the ra dius and the angle function. In Section 3 we formulate and prove the theorems on C 1 -conv erg ence for is oradial circle patterns. After a brief re view on quasicrystallic cir cle pa tterns in Section 4 we state and pr ov e in Section 5 the theor em on C ∞ -conv erg ence. The necessary results on discr ete p otential theory ar e presented in Appendix A. An extended and mor e details version of the r esults can b e found in [B ¨ uc07]. The a utho r would like to thank Alexander I. Bo b enko, B oris Spr ingb orn, and Y uri Suris for v arious disc us sions and helpful advice. 2 Circle patterns T o deﬁne circle patter ns we use combinatorial data a nd intersection ang les. The combinatorics are sp eciﬁed by a b-quad-gr aph D , that is a strongly regular cell decomp os ition of a domain in C p o s sibly with bo undary such that all 2-cells (faces) a r e em b edded and counterclo ckwise o riented. F urthermore all faces of D a re quadrilater als, that is there a r e exactly four edges incident to each face, and the 1-skeleton of D is a bipartite gr aph. W e alwa ys assume that the vertices o f D are colo r ed white and black. T o these tw o sets of vertices we asso ciate tw o planar gr aphs G and G ∗ as follows. The vertices V ( G ) ar e all white vertices of V ( D ). The edg es E ( G ) corres p o nd to faces of D , that is tw o vertices of G are co nnected by a n edge if and only if they are incident to the same face o f D . The dua l graph G ∗ is constructed analogously by taking for V ( G ∗ ) all black vertices of V ( D ). D is called simply c onne cte d if it is the cell decomp osition of a s imply connected domain of C and if every closed chain of faces is null homo topic in D . F or the intersection angles, we use a lab elling α : F ( D ) → (0 , π ) of the faces of D . By abuse of notation, α ca n also b e understo o d as a function deﬁned on E ( G ) or o n E ( G ∗ ). The lab elling α is called admissible if it satisﬁes the following condition at a ll interior black vertices v ∈ V int ( G ∗ ): X f inc ident to v α ( f ) = 2 π . (1) α Figure 1: L eft: An example of a b-qua d-graph D (black edges and bico lored vertices) and its asso c iated gra ph G (das hed edges a nd white vertices). Rig ht: The exter ior intersection ang le α of tw o intersecting circle s and the as so ciated kite built from centers and intersection p oints. 4 Ulrike B ¨ ucking (a) A regular square grid circle pattern. (b) A regular hex agonal circle pattern. (c) A rhombic embedding (a part of a Pe nr ose tiling) and a corresp onding isoradial circle pattern. Figure 2: Exa mples of isoradia l circle patterns. Deﬁnition 2.1. Let D b e a b-qua d- graph and let α : E ( G ) → (0 , π ) b e an admissible lab elling . An (immerse d plana r) cir cle p att ern fo r D (or G ) and α are a n indexed collection C = { C z : z ∈ V ( G ) } of circles in C and an indexe d collection K = { K e : e ∈ E ( G ) } = { K f : f ∈ F ( D ) } of closed kites, which a ll carry the sa me or ient ation, such that the following c o nditions ho ld. (1) If z 1 , z 2 ∈ V ( G ) are incident vertices in G , the corresp o nding cir c les C z 1 , C z 2 int e r sect with exterior int e rsection a ngle α ([ z 1 , z 2 ]). F ur thermore, the k ite K [ z 1 ,z 2 ] is bounded b y the centers o f the circles C z 1 , C z 2 , the tw o intersection po ints, and the corr esp onding edges, as in Figur e 1 (rig ht ). The intersection po ints are asso ciated to black vertices of V ( D ) o r to vertices of V ( G ∗ ). (2) If tw o fa c es a re incident in D , then the corr esp onding kites hav e o ne edg e in c ommon. (3) Let f 1 , . . . , f n ∈ F ( D ) b e the faces inciden t to an int e rior vertex v ∈ V int ( D ). Then the kites K f 1 , . . . , K f n hav e mutually disjoint in ter iors. Their unio n K f 1 ∪ · · · ∪ K f n is homeomor phic to a closed disc a nd con- tains the p o int p ( v ) corresp o nding to v in its interior. The circle pa ttern is called emb e dde d if all kites o f K have mutually disjoint int e r iors. It is ca lled isor adial if all circles of C hav e the same radius. Note that we a sso ciate a cir cle pattern C to an immers io n of the kite pattern K co rresp onding to D where the edges incident to the same white vertex are of equa l leng th. The kites can also b e reconstructed fr om the set of circ le s using the combinatorics o f G . Note further, that there are in g eneral a dditional int e r section p oints of circles which ar e not ass o ciated to black vertices of V ( D ). Some exa mples ar e shown in Figure 2 . There are also other deﬁnitions for circle patterns, for example asso ciated to a Delaunay de c omp osition o f a domain in C . This is a cell de c omp osition such that the b o undary of ea ch face is a polyg on with straight edges which is inscrib ed in a circular disk , a nd thes e disks hav e no vertices in their interior. The corres p o nding circle pattern can b e ass o ciated to the gra ph G ∗ . The Poincar´ e- dual decomp osition of a Dela unay decomp osition with the centers of the circles as vertices and straig ht edges is a Dirichlet de c omp osition (or V or onoi diagr am ) and co r resp onds to the graph G . F urthermore the deﬁnition of circle patterns can b e extended allowing cone- like s ingularities in the vertices; see [BS04] and the references therein. Circle p a tterns 5 2.1 The radius function Our study of a planar circle pattern C is based on characteriza tions and prop- erties of its radius function r C = r which assigns to every vertex z ∈ V ( G ) the radius r C ( z ) = r ( z ) o f the corresp onding circle C z . The index C will be dropp ed whenever ther e is no confusion likely . The following prop osition sp eciﬁes a necess a ry and suﬃcient co ndition for a radius function to o riginate fr om a planar circle pattern, see [BS04] for a pr o of. F or the sp ecia l ca s e of orthogo nal circle patterns with the combinatorics of the square grid, there ar e als o other character izations, see fo r example [Sch97]. Prop ositi on 2.2. L et G b e a gr aph c onstru cte d fr om a b-quad-gr aph D and let α b e an admissible lab el ling. Supp ose that C is a planar cir cle p att ern for D and α with r adius function r = r C . Then for every int erior vertex z 0 ∈ V int ( G ) we have X [ z ,z 0 ] ∈ E ( G ) f α ([ z ,z 0 ]) (log r ( z ) − log r ( z 0 )) ! − π = 0 , (2) wher e f θ ( x ) := 1 2 i log 1 − e x − iθ 1 − e x + iθ , and t he br anch of the lo garithm is chosen such that 0 < f θ ( x ) < π . Conversely, su pp ose that D is simply c onne cte d and that r : V ( G ) → (0 , ∞ ) satisﬁes (2) for every z ∈ V int ( G ) . Then ther e is a planar cir cle p attern fo r G and α whose r adius function c oincides with r . T his p attern is unique up to isometries of C . Note that 2 f α ([ z ,z 0 ]) (log r ( z ) − lo g r ( z 0 )) is the angle a t z 0 of the kite with edge lengths r ( z ) and r ( z 0 ) and angle α ([ z , z 0 ]), a s in Figure 1 (right). Equa tio n (2) is the clo s ing condition for the chain of kites co rresp o nding to the edges incident to z 0 which is co ndition (3) o f Deﬁnition 2.1. F or fur ther use we ment io n some pr o p erties of f θ , see for example [Spr03]. Lemma 2. 3. (1) The deriva tive of f θ is f ′ θ ( x ) = sin θ 2(cosh x − cos θ ) > 0 . So f θ is strictly incr e asing. (2) The function f θ satisﬁes the fun ctional e qu ation f θ ( x ) + f θ ( − x ) = π − θ . (3) F or 0 < y < π − θ the inverse function of f θ is f − 1 θ ( y ) = log sin y sin( y + θ ) . Remark 2. 4. Equation (2) can be interpreted as a nonlinear Laplace equation for the ra dius function a nd is related to a linea r discrete Laplacia n which is common in the linear theory of discr ete holomorphic functions; see for exam- ple [Duf68, Mer 01, BMS05] for more details. This can b e seen as follows. Let G b e a planar gra ph and let α be an admiss ible la be lling . Ass ume there is a smo oth one par ameter family of planar circle patterns C ε for G and α with radius function r ε for ε ∈ ( − 1 , 1). Then for every interior vertex z 0 with inciden t vertices z 1 , . . . , z m and all ε ∈ ( − 1 , 1) Pro p osition 2.2 implies that m X j =1 2 f α ([ z j ,z 0 ]) (log r ε ( z j ) − log r ε ( z 0 )) = 2 π . 6 Ulrike B ¨ ucking Diﬀerent ia ting this equatio n with resp ect to ε a t ε = 0, we o bta in m X j =1 2 f ′ α ([ z j ,z 0 ]) (log r 0 ( z j ) − log r 0 ( z 0 ))( v ( z j ) − v ( z 0 )) = 0 , (3) where v ( z ) = d dε log r ε ( z ) | ε =0 . Thus v satisﬁes a linear discrete La pla ce equation with p o s itive weights. Lemma 2 .3 and a simple calculation show that 2 f ′ α ([ z 1 ,z 2 ]) (log r 0 ( z 1 ) − log r 0 ( z 2 )) =     p ( v 1 ) − p ( v 2 ) p ( z 1 ) − p ( z 2 )     . (4) Here z 1 , z 2 ∈ V ( G ) are tw o incident vertices which cor resp ond to the c e nt e r s of circles p ( z 1 ) , p ( z 2 ) of the circle pattern C 0 . The tw o other corner p oints of the same k ite ar e denoted by p ( v 1 ) , p ( v 2 ). In analog y to smo oth harmonic functions, the radius function of a planar circle pattern satisﬁes a maximum principle a nd a Dirichlet principle. Lemma 2.5 (Maximum Principle) . L et G b e a ﬁn ite gr aph asso ciate d to a b- quad-gr aph as a b ove with some admissible lab el ling α . Supp ose C and C ∗ ar e two planar cir cle p atterns for G and α with r adius functions r C , r C ∗ : V ( G ) → (0 , ∞ ) . Then the max imu m and m inimum of t he quotient r C /r C ∗ is attaine d at the b oundary. A pro o f can b e found in [He99, Lemma 2.1]. If there exists an iso radial planar circle pattern for G and α , the usual maximum principle fo r the radius function fo llows b y taking r C ∗ ≡ 1. Theorem 2.6 (Dirichlet Principle) . L et D b e a ﬁnite simply c onne ct e d b-quad- gr aph with asso ciate d gr aph G and let α b e an admissible lab el ling. L et r : V ∂ ( G ) → (0 , ∞ ) b e some p ositive function on t he b oun dary vertic es of G . Then r c an b e extende d to V ( G ) in such a way t hat e quation (2) holds at every interior vertex z ∈ V int ( G ) if and only if ther e ex ist s any cir cle p attern for G and α . If it exists, the extension is un ique. By lack of a go o d reference we include a pro of. Pr o of. The only if part follows directly fr o m the second par t of Pro po sition 2.2. T o show the if part, a ssume that there exists a circle pattern for G and α with ra dius function R : V ( G ) → (0 , ∞ ). A function κ : V ( G ) → (0 , ∞ ) which satisﬁes the inequa lity   X [ z ,z 0 ] ∈ E ( G ) f α ([ z ,z 0 ]) (log κ ( z ) − log κ ( z 0 ))   − π ≥ 0 (5) at every interior vertex z ∈ V int ( G ) will b e c a lled su bharmonic in G. Let b be the minimu m of the quotient r /R on V ∂ ( G ) and let κ 1 be equal to r on V ∂ ( G ) and to bR on V int ( G ). Then κ 1 is clearly subhar monic. The maximu m of κ 1 /R is a ttained at the bo undary , which is a simple genera lization of the Maximum Principle 2.5. Let r ∗ be the supremum of all subharmo nic functions on G that coincide with r on V ∂ ( G ). Th us r ∗ is bo unded from ab ov e by the maximum of r/R on V ∂ ( G ) whic h is ﬁnite. One easily c he cks that r ∗ satisﬁes condition (2). The uniqueness claim follows directly from the Maximum P rinciple 2.5. Circle p a tterns 7 2.2 The angle function and relations to the radius function Similarly as for p olar co o rdinates of the c omplex plane, a suitably deﬁne d angle function can b e interpreted as a “dua l” to the r adius function. W e fo cus on con- nections betw een these functions and on a characterization for ang le functions of circle pa tter ns simila r to Prop ositio n 2.2. Let G b e a ﬁnite graph asso ciated to a b-quad-gr aph D and let α b e a n admissible labelling. Deno te by ~ E ( D ) the set of orie nt e d edges, where each edge of E ( D ) is replaced by tw o oriented edges of opp osite or ientation. Let C be a planar circle pattern for D and α . Deﬁne an angle funct ion ϕ C = ϕ on ~ E ( D ) as follows. Denote by p ( w ) the point of C corre s p o nding to the vertex w ∈ V ( D ). F or ~ e = − → z v ∈ ~ E ( D ) set ϕ ( ~ e ) = a rg( p ( v ) − p ( z )) to be the ar gument of p ( v ) − p ( z ), that is the angle b etw een the p ositively oriented real axis and the vector p ( v ) − p ( z ). As this arg ument is o nly unique up to addition of multiples of 2 π , we will mostly co nsider ϕ ∈ R / (2 π Z ). But no te that e iϕ ( ~ e ) is well deﬁned. F urthermore ϕ ( ~ e ) − ϕ ( − ~ e ) = π (mo d 2 π ) . (6) Our c ho ice of the angle function leads to the following connections betw een radius function r and angle function ϕ for a pla na r c ir cle pa ttern for D a nd α . Let f ∈ F ( D ) be a face of D . Without loss of g enerality , we assume that the no tation for the vertices and edges of f is taken from Figure 3 (left). Mor e precisely , the white vertices z − and z + of f and the black vertices v − and v + are lab elled such that the p o int s z − , v − , z + , v + app ear in this cyclical o r der using the counterclockwise o rientation of f . F urthermo re the edges are lab elled such that ~ e 1 = − − − → z − v + , ~ e 2 = − − − → z − v − , − ~ e 3 = − − − → v − z + , − ~ e 4 = − − − → v + z + . Then the fo llowing equations hold. ϕ ( ~ e 1 ) − ϕ ( − ~ e 3 ) = α ( f ) − π + 2 f α ( f ) (log r ( z + ) − log r ( z − )) (mo d 2 π ) (7) ϕ ( − ~ e 4 ) − ϕ ( ~ e 2 ) = α ( f ) − π + 2 f α ( f ) (log r ( z + ) − log r ( z − )) (mo d 2 π ) (8) ϕ ( ~ e 1 ) − ϕ ( ~ e 2 ) = 2 f α ( f ) (log r ( z + ) − log r ( z − )) (mo d 2 π ) (9) ϕ ( − ~ e 4 ) − ϕ ( − ~ e 3 ) = − 2 f α ( f ) (log r ( z − ) − log r ( z + )) (mo d 2 π ) (10) ϕ ( − ~ e 3 ) − ϕ ( ~ e 2 ) = π − α ( f ) (mo d 2 π ) (11) ϕ ( − ~ e 4 ) − ϕ ( ~ e 1 ) = α ( f ) − π (mo d 2 π ) (12) Additionally , the a ngle function ϕ satisﬁes the following Lemma 2.7 (Mono tonicity condition) . L et z ∈ V ( G ) b e a white vertex and let e 1 , . . . , e n b e the se quenc e of al l incident e dges in E ( D ) which ar e cyclic al ly or der e d r esp e cting the c ount er clo ckwise orientation of the cir cle C z . Then the values of ϕ ∈ R c an b e change d by suitably adding mu ltiples of 2 π such that ϕ ( ~ e j ) is an incr e asing fun ction of the index j and if z is an interior vertex, t hen also ϕ ( ~ e j ) − ϕ ( ~ e 1 ) < 2 π for al l j = 1 , . . . , n . The following theorem is useful to compare tw o circle pa tterns with the same combinatorics and in tersection ang les. Theorem 2. 8. L et D b e a b- quad-gr aph and let α b e an admissible lab el ling. L et C and ˆ C b e two planar cir cle p atterns for D and α with r adius functions r C = r and r ˆ C = ˆ r and angle functions ϕ C = ϕ and ϕ ˆ C = ˆ ϕ r esp e ctively. 8 Ulrike B ¨ ucking ❧ ❧ ⑤ ⑤ ✟ ✟ ✟ ✯ ✟ ✟ ✟ ✟ ❍ ❍ ❍ ❥ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❥ ❍ ❍ ❍ ❍ ✟ ✟ ✟ ✯ ✟ ✟ ✟ ✟ α α β − β + z − z + v − v + ~ e 1 ~ e 2 − ~ e 4 − ~ e 3 α 1 α n 2 β 1 r 2 2 β 2 r 1 . . . . . . r n α 2 2 β n Figure 3: L eft: A face of D with o riented edg es. R ight: An interior intersection po int with its neig hbo ring faces. Then the diﬀer en c e ˆ ϕ − ϕ gives rise to a function δ : V ( G ∗ ) → R such that the fol lowing c ondition holds on every fac e f ∈ F ( D ) . 2 f α ( f )  log  w ( z + ) w ( z − )  + log  r ( z + ) r ( z − )  − 2 f α ( f )  log  r ( z + ) r ( z − )  = δ ( v + ) − δ ( v − ) (13) Her e we have deﬁne d w : V ( G ) → R , w ( z ) = ˆ r ( z ) /r ( z ) and the n otation is taken fr om Figur e 3 (left) as ab ove. Conversely, assume that C is a planar cir cle p attern for D and α with r adius functions r C = r and angle function ϕ C = ϕ . L et δ : V ( G ∗ ) → R and w : V ( G ) → R + b e two functions which satisfy e quation (13) for every fac e f ∈ F ( D ) . T hen ( rw ) and ( ϕ + δ ) ar e t he r adius and angle fun ction of a planar cir cle p attern for D and α . This cir cle p attern is u nique up to t ra n slation. Pr o of. If C and ˆ C are t wo planar circle pa tterns for D and α , equations (6), (11), a nd (12) imply that the diﬀerence ˆ ϕ − ϕ is co nstant for all edg es incident to a ny ﬁxed black v er tex v ∈ V ( G ∗ ). Therefore δ (mo d 2 π ) is well deﬁned on the intersection p oints and enco des the relative rotation of the star of edg es a t v . Also equation (13) holds mo dulo 2 π . T o o btain a function δ with v alues in R , ﬁx δ ( v 0 ) ∈ [0 , 2 π ) for one a rbitrary vertex v 0 ∈ V ( G ∗ ) (for each connected co mp o ne nt of G ∗ ). Deﬁne the v alues of δ for all incident vertices in G ∗ by equation (13). Co ntin ue this c o nstruction until a v alue has be e n assigne d to all vertices. This pro ceedure lea ds to a well-deﬁned function, as by P rop osition 2.2 the sum of the left hand side of eq uation (13) is zero for simple closed paths in E ( G ∗ ) a round a white vertex. T o pr ov e the conv ers e cla im, observe that if r is a ra dius function, equa- tion (13) implies that ( w r ) fullﬁlls equa tion (2 ). By Prop os ition 2.2 there is a circle pattern with radius function ( wr ). Adjust the r otational freedo m at one edge according to ( ϕ + δ ). Equa tion (1 3) implies tha t ( ϕ + δ ) is indeed the a ngle function o f this circle pattern. The preceeding theore m motiv ates the de ﬁnitio n of a co mparison function for tw o circle patterns with the same combinatorics and intersection angles. Let G be a graph asso ciated to a b-quad-gr aph D and let α b e a n admiss ible lab elling. Supp ose that C 1 and C 2 are planar circle patterns for D and α with radius functions r C 1 and r C 2 and a ngle functions ϕ C 1 and ϕ C 2 resp ectively . Let δ : V ( G ∗ ) → R b e a function cor resp onding to ϕ C 2 − ϕ C 1 as in Lemma 2.8. Circle p a tterns 9 Deﬁne a c omp arison function w : V ( D ) → C by ( w ( y ) = r C 2 ( y ) /r C 1 ( y ) for y ∈ V ( G ) , w ( x ) = e iδ ( x ) ∈ S 1 for x ∈ V ( G ∗ ) . (14) Note that w ( y ) is the s c a ling factor o f the cir cle cor r esp onding to y ∈ V ( G ) when c ha nging from the circle patter n C 1 to C 2 . w ( x ) g ives the rotation o f the edge-star at x ∈ V ( G ∗ ). F urthermore, w satisﬁes the following Hir ota Equation for all fac e s f ∈ F ( D ). w ( x 0 ) w ( y 0 ) a 0 − w ( x 1 ) w ( y 0 ) a 1 − w ( x 1 ) w ( y 1 ) a 0 + w ( x 0 ) w ( y 1 ) a 1 = 0 (15) Here x 0 , x 1 ∈ V ( G ∗ ) and y 0 , y 1 ∈ V ( G ) are the black and white vertices incident to f and a 0 = x 0 − y 0 and a 1 = x 1 − y 0 are the dir ected edges. Equation (15) is the clo sing c ondition for the kite of C 2 which corres po nds to the face f . Angle functions a sso ciated to planar circle pa tterns can b e characterized in a similar wa y as radius functions are qualiﬁed in P rop osition 2 .2. Prop ositi on 2.9. L et D b e a b-quad-gr aph with asso ciate d gr aphs G and G ∗ and let α b e an admissible lab el ling. Supp ose C is a planar cir cle p attern for D and α with angle function ϕ = ϕ C . Then ϕ satisﬁes e quations (6) , (11 ) , (12) , the Monotonicity c ondition 2.7 at every white vertex of D , and the fol lowing two c onditions. (i) L et f b e a fac e of D and let e 1 and e 2 b e two e dges incident t o f and to the same white vertex and assume that e 1 and e 2 ar e enumer ate d in clo ckwise or der as in Figur e 3 (left). D eﬁne an angle β ∈ (0 , π ) by 2 β = ϕ ( ~ e 1 ) − ϕ ( ~ e 2 ) (mo d 2 π ) , wher e the orientation of the e dges is chosen s uch that the ve ctors p oint fr om a white vertex to a black vert ex , as in Figur e 4 (left). Then β + α ( f ) < π . (16) (ii) F or an int erior black vertex v ∈ V int ( G ∗ ) , denote by e 1 , . . . , e n , e n +1 = e 1 al l inci dent e dges of D in c ounter clo ckwise or der and by f j the fac e of D incident to e j and e j +1 . D enote by e ∗ j ( j = 1 , . . . , n ) the e dge incident to e j and f j which is not incident to v . F or j = 1 , . . . , n deﬁne as ab ove β j ∈ (0 , π ) by 2 β j = ϕ ( ~ e j ) − ϕ ( ~ e j ∗ ) (mo d 2 π ) , wher e we cho ose the same orientation of the e dges fr om white to black vertic es as ab ove. Then n X j =1 f − 1 α ( f j ) ( β j ) = 0 . (17) Conversely, supp ose that D is simply c onn e cte d and that ϕ : ~ E ( D ) → R / (2 π Z ) satisﬁes e quations (6) , (11) , (12) , the Monotonicity c ondition 2.7, c ondition (16) at every white vertex of D , and c ondition (17) at every int e- rior black vertex . Then ther e is a planar cir cle p attern for D and α with angle function ϕ . This p attern is unique up to sc aling and tr ans lation. 10 Ulrike B ¨ ucking Pr o of. F or a given planar circle pattern equatio ns (7)–(12) and Lemma 2.7 hold. T o show (1 6), consider a kite corres po nding to a face of D as in Figure 3 (left). Note that β − = 2 β by e quation (9), β − + β + + 2 α = 2 π , and β − , β + , α > 0. Using notation of Figure 3 (rig ht ) we also deduce that f − 1 α j ( β j ) = lo g r j +1 − log r j for j = 1 , . . . , n , wher e we identify r 1 = r n +1 . Now (17) follows immediately . In order to prove the conv er se claim, w e constr uct a radius function r : V ( G ) → (0 , ∞ ) and build a circle pattern corr esp onding to r and ϕ . Let z ∈ V ( G ) b e an interior white vertex. Set r ( z ) = 1 . Conside r a neigh- bo ring white vertex z ′ ∈ V ( G ) and the face f ∈ F ( D ) incident to z and z ′ . Denote the black vertices of D incident to z and f by v 1 , v 2 such that v 1 , z , v 2 , z ′ app ear in counterclo ckwise or der alo ng the b ounda r y of f . Deﬁne ψ − ∈ (0 , 2 π ) by ψ − = ϕ ( − → z v 1 ) − ϕ ( − → z v 2 ) (mo d 2 π ). Conditio n (16) implies that r ( z ′ ) := r ( z )exp( f − 1 α ( f ) ( ψ − / 2)) is w ell deﬁned a nd p ositive. W e pro c e ed in this wa y until a radius has be en ass igned to a ll white v er tices. Condition (17) gua r an- tees that these a ssignments do not lead to diﬀerent v alues when turning around a blac k vertex (see Figure 3 (right ) with ψ = 2 β ). Thus r is uniquely determined up to the choice of the initial radius, which co rresp onds to a globa l scaling. F o r each face f ∈ F ( D ) constr uct a kite with lengths r ( z + ) , r ( z − ) of the edges inci- dent to the white vertices z + , z − of f resp ectively and a ng le α = α ( f ). Lay out one kite ﬁxing the rotational freedom a ccording to ϕ . Successively add all other kites, resp ecting the co mbinatorics of D . By c o nstruction and as sumptions, at every interior vertex the angles o f the kites having this vertex in c o mmon add up to 2 π . Th us w e obta in a circle patter n with ang le function ϕ . 3 C 1 -con v ergence with Diric h let or Neumann b oundary con ditions In this sectio n we state and prov e o ur main res ults on conv er gence for isoradia l circle patterns. W e b eg in with Dirichlet b o unda ry conditions, that is we ﬁrst fo cus on the radius function with given b oundar y v alues. Theorem 3.1. L et D ⊂ C b e a simply c onne cte d b ounde d domain, and let W ⊂ C b e op en su ch that W c ont ains t he closur e D of D . L et g : W → C b e a lo c al ly inje ctive holomorph ic function. A ssume, for c onvenienc e, t hat 0 ∈ D . F or n ∈ N let D n b e a b-quad-gr aph with asso ciate d gr aphs G n and G ∗ n and let α n b e an admissible lab el ling. We assume that D n is simply c onne cte d and that α n is uniformly b ounde d such that for al l n ∈ N and al l fac es f ∈ F ( D n ) | α n ( f ) − π / 2 | < C (18) with some c onstant 0 < C < π / 2 indep endent of n . L et ε n ∈ (0 , ∞ ) b e a se qu en c e of p ositive num b ers such that ε n → 0 for n → ∞ . F or e ach n ∈ N , assume that ther e is an isor adia l cir cle p attern for G n and α n , wher e al l cir cles ha ve the same r adius ε n . Assume further that al l c enters of cir cles lie in the domain D and that any p oint x ∈ D which is not c ontaine d in any of t he disks b ounde d by the cir cles of the p attern has a distanc e less than ˆ C ε n to the ne ar est c ent er of a cir cle and t o the b oundary ∂ D , wher e ˆ C > 0 is some c onstant indep endent of n . Denote by R n ≡ ε n and φ n the r adius and the angle function of the ab ove cir cle p att ern for G n and α n . By abuse of C 1 -convergence for isoradial ci rcl e p a tterns 11 notation, we do n ot distinguish b etwe en t he r e alization of the cir cle p attern, that is the c ent ers of cir cles z n , the interse ction p oints v n , and t he e dges c onne cting c orr esp onding p oints in D n or G n , and t he abstr act b-qu ad-gr aph D n and t he gr aphs G n and G ∗ n . Also, the index n wil l b e dr opp e d fr om the notatio n of the vertic es and the e dges. Deﬁne another r adius function on G n as fol lows. At b oundary vertic es z ∈ V ∂ ( G n ) set r n ( z ) = R n ( z ) | g ′ ( z ) | . (19) Using The or em 2.6 ext end r n to a solution of the Dirichlet pr oblem on G n . L et z 0 ∈ V ( G n ) b e such that the disk b oun de d by the cir cle C z 0 c ontains 0 and let e = [ z 0 , v 0 ] ∈ E ( D n ) b e one of the e dges incident to z 0 such that φ n ( ~ e ) ∈ [0 , 2 π ) is minimal. L et ϕ n b e the angle function c orr esp onding to r n that satisﬁes ϕ n ( ~ e ) = a rg ( g ′ ( v 0 )) + φ n ( ~ e ) . (20) L et C n b e the planar cir cle p attern with r adius function r n and angle function ϕ n . S upp ose that C n is normalize d by a tr anslation su ch that p n ( v 0 ) = g ( v 0 ) , (21) wher e p n ( v ) denotes the interse ct ion p oint c orr esp onding to v ∈ V ( G ∗ n ) . F or z ∈ D set g n ( z ) = p n ( w ) and q n ( z ) = r n ( v ) R n ( v ) e i ( ϕ n ( − → vw ) − φ n ( − → vw )) , wher e w is a vertex of V ( G ∗ n ) closest to z and v is a vertex of V ( G n ) closest to z such that [ v , w ] ∈ E ( D n ) . Then q n → g ′ and g n → g uniformly on c omp act subsets in D as n → ∞ . Remark 3.2. The pro of of Theo rem 3.1 actually shows the following a prior i estimations for the a ppr oximating functions q n and g n . k q n − g ′ k V ( G n ) ∩ K ≤ C 1 ( − log 2 ε n ) − 1 2 and k g n − g k V ( G n ) ∩ K ≤ C 2 ( − log 2 ε n ) − 1 2 for all co mpact s e ts K ⊂ D , where the constant s C 1 , C 2 depe nd on K , g , D , and on the constants of Theorem 3 .1. W e b eg in with an a priori estimation for the q uotients of the radius functions. Lemma 3. 3. F or z ∈ V ( G n ) set h n ( z ) = log | g ′ ( z ) | , t n ( z ) = log ( r n ( z ) / R n ( z )) = log ( r n ( z ) / ε n ) . Then h n ( z ) − t n ( z ) = O ( ε n ) . Here and b elow the no ta tion s 1 = O ( s 2 ) means that there is a constant C which ma y dep end o n W, D , g , but not on n and z , such that | s 1 | ≤ C s 2 wherever s 1 is deﬁned. A dir ect co ns equence o f L e mma 3.3 is r n ( z ) = R n ( z ) | g ′ ( z ) | + O ( ε 2 n ) . (22) Our pro o f us es ideas of Sc hr amm’s pro of of the co rresp onding Lemma in [Sch97]. 12 Ulrike B ¨ ucking Pr o of. Cons ider the function p ( z ) = t n ( z ) − h n ( z ) + β | z | 2 , where β ∈ (0 , 1) is some function o f ε n . W e wan t to cho ose β such that p will hav e no maximum in V int ( G n ). Suppo se tha t p ha s a maximum a t z ∈ V int ( G n ). Denote by z 1 , . . . , z m the incident vertices o f z in G n in c ounterclockwise o rder. Then we have t n ( z j ) − t n ( z ) ≤ x j (23) for j = 1 , . . . , m wher e x j = h n ( z j ) − h n ( z ) − β | z j | 2 + β | z | 2 . (24) Since z ∈ V int ( G n ), we hav e | z | = O (1 ) and by a s sumption z − z j = O ( ε n ). With β ∈ (0 , 1) this lea ds to β | z j | 2 − β | z | 2 = O ( ε n ). U sing this estimate and the s mo othness of Re(log g ′ ), we get x j = O ( ε n ). F rom (23), the deﬁnition of t n ( z ) = lo g( r n ( z ) / R n ( z )) = lo g r n ( z ) − lo g ε n and the monotonicity of the sum in equation (2) (see Lemma 2.3 (i)), we get 0 =    m X j =1 f α ( z ,z j ) (log r n ( z j ) − log r n ( z ) | {z } = t n ( z j ) − t n ( z ) )    − π ≤   m X j =1 f α ( z ,z j ) ( x j )   − π . (25) Remem ber ing x j = O ( ε n ), we can consider a T aylor expansion o f the right hand side of inequality (2 5) ab o ut 0 in order to ma ke a n O ( ε 3 n )-analysis. Consider the chain of faces f j of D n ( j = ✐ ✐ ② ② ✟ ✟ ✯ ✟ ✟ ✟ ✟ ❍ ❍ ❥ ❍ ❍ ❍ ❍ ❍ ❍ ❥ ❍ ❍ ❍ ❍ ✟ ✟ ✯ ✟ ✟ ✟ ✟ α j α j z z j v j − 1 v j a j a j − 1 a j − 1 a j Figure 4: A rho m bic face of D n with o riented edges . 1 , . . . , m ) which are incident to z and z j . The enum e ration of the vertices z j (and hence of the faces f j ) a nd of the black vertices v 1 , . . . , v m in- cident to these faces ca n b e chosen such that f j is incident to v j − 1 and v j for j = 1 , . . . , m , wher e v 0 = v m . F urthermore, using this enumeration i ( z j − z ) and ( v j − v j − 1 ) are parallel, s ee Figure 4. As each face f j of an isoradial c ir cle pattern is a r hombus w e can write, using the no tation of Figure 4, z j − z = a j − 1 + a j and v j − v j − 1 = a j − a j − 1 . (26) Denoting α j = α ([ z , z j ]), l j = | z j − z | = 2 ε n sin( α j / 2), and ˆ l j = | v j − v j − 1 | = 2 ε n cos( α j / 2), we eas ily obta in by simple c alculations that f α j (0) = ( π − α j ) / 2 , f ′ α j (0) = ˆ l j / (2 l j ) , f ′′ α j (0) = 0 . T aking in to account that equation (2) holds with R n ≡ ε n and using the uniform bo undedness (1 8 ) of the lab elling α , inequality (25) yields 0 ≤ m X j =1 f ′ α j (0) x j + O ( ε 3 n ) . (27) C 1 -convergence for isoradial ci rcl e p a tterns 13 T o ev aluate this sum, expand h n ( z j ) − h n ( z ) = Re(log g ′ ( z j ) − log g ′ ( z )) = Re( a ( z j − z ) + b ( z j − z ) 2 ) + O ( ε 3 n ) and x j = h n ( z j ) − h n ( z ) − β | z j | 2 + β | z | 2 = Re( a ( z j − z ) + b ( z j − z ) 2 − 2 β ¯ z ( z j − z )) − β l 2 j + O ( ε 3 n ) . Noting that f ′ α j (0)( z j − z ) = ( v j − v j − 1 ) / (2 i ) we get m X j =1 f ′ α j (0) x j = Re a − 2 β ¯ z 2 i m X j =1 ( v j − v j − 1 ) | {z } =0 + b 2 i m X j =1 ( v j − v j − 1 )( z j − z ) | {z } (26) = ( a j − a j − 1 )( a j + a j − 1 ) | {z } =0 ! − β m X j =1 l j ˆ l j 2 + O ( ε 3 n ) . Thu s fro m ineq uality (27), remem b er ing l j = 2 ε n sin( α j / 2) and ˆ l j = 2 ε n cos( α j / 2), we arrive a t 0 ≤ − β ε 2 n m X j =1 sin( α j / 2) cos( α j / 2) + O ( ε 3 n ) ⇐ ⇒ β m X j =1 sin( α j ) ≤ O ( ε n ) . Note that ε 2 n P m j =1 sin( α j ) > π ε 2 n is the a rea of the rhombic faces incident to the vertex z . Thus we co nclude that β = O ( ε n ). This means, that if we choose β = C ε n with C > 0 a suﬃcie ntly large constant and if ε n is s ma ll eno ugh s uch that C ε n < 1, then p will have no maximum in V int ( G n ). In that case, as we hav e p ( z ) = β | z | 2 = O ( ε n ) on V ∂ ( G n ), we deduce that p ( z ) ≤ O ( ε n ) in V ( G n ) and th us t n ( z ) − h n ( z ) ≤ O ( ε n ) for z ∈ V ( G n ) . (28) The pro of for the reverse inequality is a lmost the same. The only modiﬁ- cations needed a re r eversing the sign of β and a few inequalities. Remark 3. 4. The statement of Le mma 3.3 can be improved to h n ( z ) − t n ( z ) = O ( ε 2 n ) (29) in the cas e of a ’very regular ’ isoradial circle pattern. These are iso radial circle patterns such that for every o r iented edge e j 1 = z j 1 − z ∈ ~ E ( G ) incident to an int e r ior vertex z ∈ V int ( G ) there is another pa rallel edge e j 2 = z j 2 − z ∈ ~ E ( G ) with opposite direction incident to z , tha t is e j 2 = − e j 1 . F urthermore, the corres p o nding in ters e c tion angles agree: α ([ z , z j 1 ]) = α ([ z , z j 2 ]). This additional regular ity prop erty holds for example fo r an ortho g onal circle pattern with the combinatorics of a par t of the square g rid, s e e Figur e 2(a). The proo f o f estimation (29) follows the sa me reasonings a s ab ove, but makes an O ( ε 4 n )-analysis. The additiona l regularity implies that all terms of order ε 3 n v anish. Deﬁnition 3. 5. F or a function η : V ( G ) → R deﬁne a discrete Laplacian by ∆ η ( z ) = X [ z ,z j ] ∈ E ( G ) 2 f ′ α ([ z ,z j ]) (0)( η ( z j ) − η ( z )) . (30) 14 Ulrike B ¨ ucking As f ′ α ([ z 1 ,z 2 ]) (0) > 0 one immediately has the following Lemma 3.6 (Maximum Principle) . If ∆ η ≥ 0 on V int ( G ) then the maximum of η is attaine d at the b oun dary V ∂ ( G ) . The proof of Lemma 3.3 actually shows, that t n − h n is almo st har monic. More precisely , w e have ∆( t n − h n ) = O ( ε 3 n ). Adding a s uita ble subharmonic function β | z | 2 with β > 0 big enough, we deduce that the resulting function p is subharmonic , that is ∆ p ≥ 0, such that p atta ins its maximum at the bo undary . This is also impo r tant for our pr o of o f the following lemma. Lemma 3.7. L et t n and h n b e deﬁne d as in L emma 3.3. L et K ⊂ D b e a c omp act subset in D . Then the fol lowing est imation holds for every interior vertex z ∈ V int ( G n ) ∩ K su ch that al l its incident vertic es z 1 , . . . , z l ar e also in V int ( G n ) ∩ K : t n ( z j ) − h n ( z j ) − ( t n ( z ) − h n ( z )) = O ( ε n ( − log ε n ) − 1 2 ) . (31) for j = 1 , . . . , l . The c onstant in the O - notation may dep en d on K , but not on n or z . The pro of of Lemma 3.7 uses the following estimation for sup erhar monic functions, which is a version o f Cor ollary 3.1 of [SC97]; see also [SC97, Re- mark 3.2 and Lemma 2 .1]. Prop ositi on 3.8 ([SC97]) . L et G b e an un dir e cte d c onne cte d gr aph without lo ops and let c : E ( G ) → R + b e a p ositive weight function on the e dges. Denote c ( e ) = c ( x, y ) for an e dge e = [ x, y ] ∈ E ( G ) and assume that m = max [ x,y ] ∈ E ( G ) X [ x,z ] ∈ E ( G ) c ( x, z ) c ( x, y ) < ∞ . Denote by d ( x, y ) the c ombinatorial distanc e b et we en two vertic es x, y ∈ V ( G ) in the gr aph G . L et B x (  ) = { y ∈ V ( G ) : d ( x, y ) ≤  } b e the c ombinatorial b al l of r adius  > 0 ar ound the vertex x ∈ V ( G ) . Fix x ∈ V ( G ) and R ≥ 4 and set A R = sup 1 ≤  ≤ R  − 2 W x (  ) , wh ere W x (  ) = X z ∈ B x (  ) , y ∈ V ( G ) d ( x,z ) 0 and ε n small enough we deduce similar ly as in the pro of of Lemma 3.3 that ∆ p ( z ) ≥ 0 for a ll interior vertices z ∈ V int ( G n ). Now deﬁne the po sitive function ˆ p = ε n + k p k − p . Then ∆ ˆ p ( z ) ≤ 0 for all z ∈ V int ( G n ). The pro of of Lemma 3.3 shows that there is a co nstant C 1 , dep e nding only on g , D , and the la be lling α , such that k p k ≤ C 1 ε n . Thus k ˆ p k ≤ C 2 ε n with C 2 = 2 C 1 + 1. T o ﬁnish to pro o f, we apply Prop osition 3.8 to the sup er harmonic function ˆ p . Remem ber that G n is a co nnected graph without lo ops and c ( e ) := 2 f ′ α ( e ) (0) > 0 deﬁnes a p ositive weight function on the edges. The b ound (1 8) o n the lab elling α implies that m = max [ x,y ] ∈ E ( G n ) X [ x,z ] ∈ E ( G n ) c ( x, z ) c ( x, y ) < 2 π π / 2 − C cot( π / 4 − C / 2) cot( π / 4 + C / 2) =: C 3 < ∞ . Let x ∈ V int ( G n ). Note that W x (  ) = X z ∈ B x (  ) , y ∈ V ( G n ) d ( x,z ) ε 2 n sin( π / 2 − C ) is the ar ea of the face f ∈ F ( D n ). Thus | F w ( x,  ) | < π ((  + 1 )2 ε n ) 2 ε 2 n sin( π / 2 − C ) ≤ 16 π  2 sin( π / 2 − C ) =:  2 C 4 for all  ≥ 1. Therefor e we o btain A R = sup 1 ≤  ≤ R  − 2 W x (  ) < C 4 , wher e the upper b o und C 4 is independent o f R ≥ 4 and n ∈ N . Let K ⊂ D be co mpact. Denote b y e the Euclidean distance (b etw een a po int and a compact set or b etw ee n closed sets of R 2 ∼ = C ). Let z ∈ V ( G n ) ∩ K and set ( R + 1) = d ( z , V ∂ ( G n )) to b e the combinatorial distance from z to the boundar y o f G n . Let z j ∈ V ( G n ) b e incident to z . As the lab elling α is bo unded, ε n → 0, a nd D n approximates D , we deduce that R ≥ ε − 1 n C 5 ≥ 4 if n ≥ n 0 is lar ge enough. Th us for a ll n ≥ n 0 and all z ∈ V ( G n ) ∩ K 1 / p log 2 R ≤ 1 / p log 2 C 5 − log 2 ε n ≤ √ 2 / p − log 2 ε n holds by o ur a ssumptions. Prop os ition 3.8 implies that     ˆ p ( z ) ˆ p ( z j ) − 1     ≤ 4 C 2 3 √ C 4 √ 2 p − c ( z , z j ) log 2 ε n for a ll inciden t vertices z , z j ∈ V ( G n ) ∩ K and n ≥ n 0 . As c ( z , z j ) ≤ co t( π/ 4 − C / 2 ) / 2 and k ˆ p k ≤ C 2 ε n we ﬁnally arr ive at the des ired es timation | t n ( z j ) − h n ( z j ) − ( t n ( z ) − h n ( z )) | = | ˆ p ( z ) − ˆ p ( z j ) | ≤ C 6 ε n ( − log 2 ε n ) − 1 2 16 Ulrike B ¨ ucking for all incident vertices z , z j ∈ V ( G n ) ∩ K and n ≥ n 0 , wher e the constant C 6 depe nds o n C 2 , . . . , C 5 , that is only on g , D , C , ˆ C , and K . Lemma 3.9. L et ~ e = − → uv ∈ ~ E ( D n ) b e a dir e cte d e dge with u ∈ V ( G n ) and v ∈ V ( G ∗ n ) . D enote by δ n ( e ) the c ombinatorial distanc e in D n fr om e = [ u, v ] to [ z 0 , v 0 ] , that is the le ast int e ger k such that ther e is a se quenc e of e dges { [ z 0 , v 0 ] = e 1 , e 2 , . . . , e k = e } ⊂ E ( D n ) such t hat the e dges e m +1 and e m ar e incident to the same fac e in D n for m = 1 , . . . , k − 1 . Then ϕ n ( ~ e ) = a rg g ′ ( v ) + φ n ( ~ e ) + δ n ( e ) O ( ε n ( − log 2 ε n ) − 1 2 ) . (32) The c onstant in the notation O ( ε n ( − log 2 ε n ) − 1 2 ) may dep end on t he distanc e of v to the b oun dary ∂ D . Note that if ∂ D is smo o th, then δ n ( e ) = O ( ε − 1 n ). In g eneral w e have δ n ( e ) = O ( ε − 1 n ) on compact subsets K ⊂ D , where the constant in the notatio n O ( ε − 1 n ) may depend on K . In any ca se, o n compact subsets of D we hav e ϕ n ( ~ e ) = arg g ′ ( v ) + φ n ( ~ e ) + O (( − log 2 ε n ) − 1 2 ) . (33) Pr o of. Using the notation o f Fig ure 3 (left), equation (31) implies f α (log r n ( z + ) − log r n ( z − )) = f α ( t n ( z + ) − t n ( z − )) = f α (0) + f ′ α (0)(log | g ′ ( z + ) | − log | g ′ ( z − ) | ) + O ( ε n ( − log 2 ε n ) − 1 2 ) . As in Lemma 3.3 we hav e 2 f ′ α (0) = | v + − v − | | z + − z − | = v + − v − i ( z + − z − ) with the same notatio n which yields 2 f ′ α (0)(log | g ′ ( z + ) | − log | g ′ ( z − ) | ) = v + − v − i ( z + − z − ) Re( a ( z + − z − )) + O ( ε n ( − log 2 ε n ) − 1 2 ) = Im( a ( v + − v − )) + O ( ε n ( − log 2 ε n ) − 1 2 ) = ar g g ′ ( v + ) − ar g g ′ ( v − ) + O ( ε n ( − log 2 ε n ) − 1 2 ) , where a = g ′′ (( z + + z − ) / 2) g ′ (( z + + z − ) / 2) = g ′′ (( v + + v − ) / 2) g ′ (( v + + v − ) / 2) . By Lemma 2.7 w e ca n choose the angle functions φ n and ϕ n on any mini- mal sequence o f edges { [ z 0 , v 0 ] = e 1 , e 2 , . . . , e k = e } ⊂ E ( D n ) such that equa- tions (7)–(1 2) are s a tisﬁed without the (mo d 2 π )-term. Using the a b ove co nsid- erations o f 2 f α (log r n ( z + ) − log r n ( z − )) and the norma liz ation o f ϕ n , w e arrive at equa tion (32 ). Pr o of of The or em 3.1. Consider a compact subset K of D . Let z ∈ V ( G n ) ∩ K and v ∈ V ( G ∗ n ) ∩ K be v er tices which are incident in D n , that is [ z , v ] ∈ E ( D n ). Then Lemmas 3.3 and 3.9 imply that log g ′ ( z ) = log | g ′ ( z ) | + i a rg g ′ ( z ) = log( r n ( z ) / R n ( z )) + i ( ϕ n ( − → z v ) − φ n ( − → z v )) + O (( − log 2 ε n ) − 1 2 ) . C 1 -convergence for isoradial ci rcl e p a tterns 17 As g ′ and thus the quotient r n /R n is uniformly b ounded, we obta in g ′ ( z ) = r n ( z ) R n ( z ) e i ( ϕ n ( − → z v ) − φ n ( − → z v )) | {z } = q n ( z ) + O (( − log 2 ε n ) − 1 2 ) . (34) This implies the uniform conv erg ence on compact subse ts of D o f q n to g ′ . Conv erg e nce o f g n is now pr ov en by using suitable integrations of g ′ and q n . Let w ∈ V ( G ∗ n ) and consider a shortest path γ in G ∗ n from v 0 to w with vertices { v 0 = w 1 , w 2 , . . . , w k = w } ⊂ V ( G ∗ n ). Then g ( w ) = g ( v 0 ) + Z γ g ′ ( ζ ) dζ = g ( v 0 ) + k − 1 X j =1 g ′ ( w j +1 )( w j +1 − w j ) + O ( ε n ) = g ( v 0 ) + k − 1 X j =1 q n ( w j +1 )( w j +1 − w j ) + O (( − log 2 ε n ) − 1 2 ) , bec ause g ′ ( w j ) − q n ( w j ) = O (( − log 2 ε n ) − 1 2 ), w j +1 − w j = O ( ε n ) a nd k = O ( ε − 1 n ) on compact s e ts. Thus it only remains to show that p n ( w ) = g ( v 0 ) + k − 1 X j =1 q n ( w j +1 )( w j +1 − w j ) + O (( − log 2 ε n ) − 1 2 ) . (35) Remem ber ing q n ( v + ) = r n ( z + ) R n ( z + ) e i ( ϕ n ( − − − − − → w j +1 z + ) − φ n ( − − − − − → w j +1 z + )) + O (( − log 2 ε n ) − 1 2 ) , ( w j +1 − w j ) = 2 R n ( z + ) cos( α ([ w j +1 , w j ]) / 2)e i ( φ n ( − − − − − → w j +1 z + ) − ( π / 2 − α ([ w j +1 ,w j ]) / 2)) , we can c onclude that q n ( w j +1 )( w j +1 − w j ) = p n ( w j +1 ) − p n ( w j ) + O ( ε n ( − log 2 ε n ) − 1 2 ) , where z − , z + ∈ V ( G ) are incident to w j +1 and w j and we hav e used the no tations are as in Figure 3 (left) w ith w j = v − and w j +1 = v + . As we have normaliz e d g ( v 0 ) = p ( v 0 ), this prov es equation (35) a nd ther efore the unifor m c o nv erge nc e of p n to g on compact subsets of D . Remark 3.10. Theo r em 3 .1 may e a sily be generalized in the following wa ys. First, we may cons ider ’nearly isora dial’ circle patterns whic h satisfy R n ( z ) = O ( ε n ) for all vertices z ∈ V ( G n ) and R n ( z 1 ) /R n ( z 2 ) = 1 + O ( ε 3 n ) for all edges [ z 1 , z 2 ] ∈ E ( G n ). Second, we may omit the a ssumption that the whole doma in D is approx- imated b y the rhom bic em b eddings D n . Then the con vergence c laims r emain true for co mpact subs e ts of any op en doma in D ′ ⊂ D whic h is covered or ap- proximated by the r hombic e m be dding s and contains v 0 . Using the angle function instead of the radius function, we obtain the fol- lowing analog of Theo rem 3.1 for Neumann b ounda r y co nditions. 18 Ulrike B ¨ ucking Theorem 3. 11. Under the s ame assumptions as in The or em 3.1 and with t he same notation, assume further that ε n is suﬃciently smal l such that for al l n ∈ N sup v ∈ D max θ ∈ [0 , 2 π ] | arg g ′ ( v + 2 ε n e iθ ) − arg g ′ ( v ) | < π 2 − C < min e ∈ E ( G n ) ( π − α ( e )) . (36) Deﬁne an angle function on t he oriente d b oundary e dges by ϕ n ( ~ e ) = φ n ( ~ e ) + ar g g ′ ( v ) , wher e ~ e = − → z v ∈ ~ E ( D n ) and v ∈ V ( G ∗ n ) . The n t her e is a cir cle p attern C n for G n and α n with ra dius fun ction r n and angle function ϕ n with t hese b ou n dary values. Supp ose that this cir cle p attern is n ormalize d su ch that r n ( z 0 ) = R n ( z 0 ) | g ′ ( z 0 ) | , wher e z 0 ∈ V ( G n ) is chosen such that the disk b ou n de d by the cir cle C z 0 c ontains 0 . Su pp ose further t hat C n is normalize d by a tr anslation such that p n ( v 0 ) = g ( v 0 ) , (37) wher e p n ( v ) denotes the interse ction p oint c orr esp onding to v ∈ V ( G ∗ n ) . Then q n → g ′ and g n → g uniformly on c omp act su bsets in D as n → ∞ . Pr o of. The ex is tence claim for the cir cle pattern with Neumann bo undary con- ditions follows fro m [BS04, Theore m 3] us ing the assumption (36). Theorem 2.8 shows that the diﬀerence ϕ n − φ n gives ris e to a function δ n : V ( G ∗ n ) → R with b oundar y v alues given by arg g ′ . The pro of of the convergence claim is similar to the pro o f of Theorem 3 .1. The roles of δ n = ϕ n − φ n and log( r n /R n ) hav e to b e interch anged in Lemmas 3 .3, 3 .7, and 3.9 and s imilarly arg g ′ = Im(log g ′ ) has to be conside r ed instead of lo g | g ′ | = Re(lo g g ′ ). The r ole of equa tio n (2) is substituted b y equation (17). 4 Quasicrystallic circle patterns The order of con vergence in Theorems 3.1 and 3.11 can be improved for a sp ecial class o f isor adial circle pa tterns with a unifo r mly b o unded num b er of diﬀerent edge directions and a lo cal deformation pro p er ty . In the following, we introduce suitable ter mino logy and so me useful results. As the kites of a n iso radial cir cle pattern a re in fact rhombi, an embedded isoradia l cir cle pattern leads to a rhombic emb e dding in C of the cor resp onding b-quad-gr aph D . Conversely , adding circles with centers in the white vertices o f a r ho mbic em b edding and r adius eq ual to the edge length results in an em b edded isoradia l circle pattern. Given a r hombic embedding of a b- quad-gr a ph D , consider for each dire c ted edge ~ e ∈ ~ E ( D ) the vector of its embedding as a co mplex n um b er with length one. Half of the num b er o f diﬀerent v alues of these directions is called the dimension d of the rhom bic embedding. If d is ﬁnite, the rhombic embedding is calle d quasicrystal lic . A circle pattern for a b-quad-g raph D is ca lled a quasicrystal lic cir cle p attern if ther e ex is ts a q uasicrysta llic rhombic embedding of D and if the Quasicr yst allic circle p a tterns 19 int e r section angles ar e taken from this rho mb ic embedding. The compa rison function of the isoradial circle pattern C 1 for D and the quasicrys tallic cir cle pattern C 2 will also b e called c omp arison fun ction for C 2 . Quasicry s tallic c ircle patterns w ere intro duced in [BMS0 5]. Certainly , this prop erty o nly makes sense for inﬁnite gr aphs or inﬁnite sequences of gr aphs with growing num b er of vertices and edges. In the following we will identify the b-qua d-graph D with a rhombic embed- ding of D . 4.1 Quasicrystallic rhombic embeddings and Z d An y rhombic embedding of a connected Figure 5: An example of a combinatorial surfa c e Ω D ⊂ Z 3 . b-quad-gr aph D can be s een as a sort of pro- jection of a cer tain tw o-dimensio na l sub co m- plex (combinatorial surface) Ω D of the m ulti- dimensional lattice Z d (or of a m ulti-dimen- sional lattice L which is isomorphic to Z d ). An illustr ating e x ample is given in Figure 5 . The combinatorial surface Ω D in Z d can be constructed in the following wa y . Denote the set of the diﬀeren t edge directions of D by A = { ± a 1 , . . . , ± a d } ⊂ S 1 . W e supp os e that d > 1 and tha t a ny tw o non-o pp o site el- ement s of A are linearly indep endent over R . Let e 1 , . . . , e d denote the standar d or thonor- mal ba sis of R d . Fix a white vertex x 0 ∈ V ( D ) and the origin of R d . Add the edges of {± e 1 , . . . , ± e d } at the origin whic h cor r esp ond to the edges of {± a 1 , . . . , ± a d } incident to x 0 in D , tog ether with their endp oints. Successively contin ue the construction at the new endpoints. Also, add tw o- dimens io nal facets (faces) of Z d corres p o nding to face s of D , spanned by incident edges. A combinatorial sur face Ω D in Z d corres p o nding to a quasicrys ta llic rho m- bic em b edding can b e c haracteriz ed using the following mono tonicity prop erty , see [BMS05, Section 6] for a pr o of. Lemma 4.1 (Monoto nicity c r iterium) . Any t wo p oints of Ω D c an b e c onne cte d by a p ath in Ω D with al l dir e cte d e dges lying in one d -dimensional o ctant, that is al l dir e cte d e dges of this p ath ar e elements of one of the 2 d subsets of {± e 1 , . . . , ± e d } c ontaining d line arly indep endent ve ctors. An impo rtant class of examples of rho mb ic embeddings of b-q uadgraphs c a n be co nstructed using ideas of the grid pro jection metho d for quasip erio dic tilings of the plane ; see for example [DK8 5, GR86, Sen95]. Example 4 .2 (Quasic r ystallic rho mbic embedding obta ined fr o m a pla ne) . Let E be a t wo-dimensional plane in R d and t ∈ E . L e t e 1 , . . . , e d be the s ta ndard orthonor mal ba sis of R d . W e a s sume that E do es not contain any of the seg- men ts s j = { t + λ e j : λ ∈ [0 , 1] } for j = 1 , . . . , d . Then we can choose p ositive nu m b er s c 1 , . . . , c d such that the or tho gonal pr o jections P E ( c j e j ) have length 1. (If E c ontains t wo diﬀerent segments s j 1 and s j 2 , the following construc- tion only leads to the sta ndard squa r e grid pa ttern Z 2 . If E contains ex actly one s uch segment s j , then the co nstruction may b e adapted for the r emaining 20 Ulrike B ¨ ucking E Ω L ( E ) (a) Straight l ine E . E △ Ω L ( E △ ) (b) Mo diﬁed line E △ with a “hat”. Figure 6: Example for the usage of the grid-pro jectio n metho d in Z 2 . The V oronoi cells which contain Ω L are mar ked in red. dimensions e xcluding e j .) W e further assume that the ortho gonal pr o jections onto E of the tw o -dimensional facets E j 1 ,j 2 = { λ 1 e j 1 + λ 2 e j 2 : λ 1 , λ 2 ∈ [0 , 1] } for 1 ≤ j 1 < j 2 ≤ d are non-degener ate paralle lo grams. Consider around each v e rtex p of the lattice L = c 1 Z × · · · × c d Z the hypercub oid V = [ − c 1 / 2 , c 1 / 2] × · · · × [ − c d / 2 , c d / 2], that is the V orono i cell p + V . These trans lations of V co ver R d . W e build an inﬁnite monotone tw o- dimensional surface Ω L ( E ) in L by the following constr uction. The basic idea is illustra ted in Figur e 6 (left). If E in tersects the interior of the V oronoi cell of a lattice po int (i.e. ( p + V ) ◦ ∩ E 6 = ∅ for p ∈ L ), then this point b elongs to Ω L ( E ). Undirected e dges corres p o nd to intersections o f E with the interior of a ( d − 1)-dimensional facet bo unding t wo V oronoi cells. Thus w e get a connected graph in L . An inter- section of E with the interior o f a tr anslated ( d − 2)-dimensiona l facet of V corres p o nds to a recta ngular tw o-dimensio nal face of the la ttice. By cons truc- tion, the o rthogona l pro jection of this graph onto E re s ults in a plana r connected graph who se faces are a ll of even degree (= num b er o f incident edges o r of in- cident vertices). A face of degr e e bigger than 4 co rresp onds to an intersection of E with the translation o f a ( d − k )-dimensional facet of V for so me k ≥ 3. Consider the vertices a nd edges of such a fac e a nd the cor resp onding po ints and edges in the lattice L . These points lie o n a combinatorial k -dimensional hypercub oid cont a ined in L . By constr uction, it is easy to s ee that there are t wo p oints of the k -dimensional hyperc ub o id which are each inciden t to k of the g iven vertices. W e choose a p oint with lea st distance fro m E and a dd it to the surface. Adding edges to neigh b o ring vertices splits the face of degree 2 k int o k fa c es of degree 4. Th us we obtain an inﬁnite monoto ne t wo-dimensional combinatorial surface Ω L ( E ) which pro jects to an inﬁnite rhombic embedding cov ering the whole plane E . Example 4.3 (Mo diﬁcation of the construction in Example 4.2) . The metho d used in the preceding example can b e mo diﬁed to result in similar but diﬀerent rhombic embeddings. T he bas ic idea is illustrated in Figur e 6 (right). Let E b e a tw o-dimensiona l plane in R d satisfying the sa me ass umptions as in Example 4 .2. Let N 6 = 0 be a vector orthogo nal to E . Let △ be an equilatera l triangle in E with v er tice s t 1 , t 2 , t 3 and let s b e the in ter section p o int of the bisecting lines o f the a ngles. Co nsider the tw o -dimensional facets o f the thre e- Quasicr yst allic circle p a tterns 21 dimensional tetrahedron T ( △ , N ) spanned b y the four vertices t 1 , t 2 , t 3 , s + N . Exactly o ne of these facets is completely c o ntained in E (this is the tria ngle △ ). W e remove the triangle from E and add instead the remaining facets of T ( △ , N ). Let E △ be the resulting t wo-dimensional sur face. No te that E △ is orientable like E . Deﬁne γ ∈ (0 , π / 2) by γ = arctan(2 √ 3 k N k / k t 1 − t 2 k ), wher e k · k denotes the Euclidean norm of v ec tors in R d . γ is the acute ang le betw een E a nd the tw o -dimensional facets of T ( △ , N ) not contained in E . If γ is small e no ugh, then our assumptions imply tha t we can apply the same construction algo r ithm to E △ as for the plane E in the prev ious example and obtain a mono tone surfac e Ω L ( E △ ). The orthogo na l pro jection o f Ω L ( E △ ) onto E is a rhombic embedding whic h coincides w ith the rho mb ic em b edding from Ω L ( E ) except for a ﬁnite pa rt. The quasicrystallic rhombic e mbedding o f Example 4.3 may also o bta ined using the following gener al concept of lo cal changes of rho m bic embeddings. Deﬁnition 4. 4. Let D be a rhombic embedding of a ﬁnite simply connected b-quad-gr aph with corresp o nding combinatorial sur face Ω D in Z d . Let ˆ z ∈ V int (Ω D ) b e an interior v er tex Figure 7: A ﬂ ip of a three- dimensional cub e . The dashed edges and their inciden t faces are not par t of the surface in Z d . with exactly three incident tw o-dimensio nal facets o f Ω D . Consider the three-dimensio- nal cub e with these bo undary face ts. Re- place the three given facets with the three other t wo-dimensional fa c e ts o f this cub e. This pr o cedure is called a ﬂip ; s ee Figure 7 for an illustr ation. A vertex z ∈ Z d c an b e r e ache d with ﬂips fr om Ω D if z is contained in a combinatorial surface obta ine d from Ω D by a suitable se- quence of ﬂips. The s et o f a ll vertices which can b e r eached with ﬂips (including V (Ω D )) will b e denoted by F (Ω D ). F or further use we enlar ge the set F (Ω D ) of vertices which can b e reached by ﬂips from Ω D in the following way . F or a s et of vertices W ⊂ V ( Z d ) deno te by W [1] the set W to g ether with all vertices incident to a t wo-dimensional facet of Z d where three of its four v ertices b elong to W . Deﬁne W [ k +1] = ( W [ k ] ) [1] inductively for all k ∈ N . In par ticular, we deno te for s ome arbitra ry , but ﬁxed κ ∈ N F κ (Ω D ) = ( F (Ω D )) [ κ ] . 4.2 Quasicrystallic circ le patterns and in tegrabilit y Let D be a quasicry s tallic rhombic e mbedding of a b- q uad-gra ph. The combi- natorial surface Ω D in Z d is imp ortant by its connection with integrability . See also [BS0 8] for a more deta iled pres entation and a deepened study of integra- bilit y a nd cons istency . In par ticular, a function deﬁned on the vertices of Ω D which satisﬁes some 3D-consistent e quation on all faces of Ω D can uniquely b e e xtended to the brick Π(Ω D ) := { n = ( n 1 , . . . , n d ) ∈ Z d : a k (Ω D ) ≤ n k ≤ b k (Ω D ) , k = 1 , . . . , d } , 22 Ulrike B ¨ ucking where a k (Ω D ) = min n ∈ V (Ω D ) n k and b k (Ω D ) = max n ∈ V (Ω D ) n k . Note that Π(Ω D ) is the hull of Ω D . A pr o of may b e found in [BMS05, Se c tio n 6]. This ex- tension of a function using a 3 D-consistent equation will now b e applied for the compariso n function w deﬁned in (1 4) of tw o cir c le patterns. In pa rticular, we take for C 1 the isor adial circle pattern whic h corr e sp onds to the qua sicrystallic rhombic embedding D . Given ano ther cir cle pa ttern C 2 for D with the same int e r section angles, let w b e the co mparison function for C 2 . Note that the Hi- rota Equation (15) is 3 D-consistent; see Sectio ns 10 and 11 of [BMS05] for mo re details. Thus w considered a s a function on V (Ω D ) can uniquely b e extended to the brick Π(Ω D ) such that equa tion (15) holds o n a ll tw o-dimensiona l facets . Additionally , w and its e x tension are r e al v alued on white p oints V w (Ω D ) and has v alue in S 1 for black p o ints V b (Ω D ). This ca n easily b e deduced from the Hirota Equation (15). The extension o f w can be used to deﬁne a ra dius function for any rhombic embedding with the same b o undary faces a s D . Lemma 4.5. L et D and D ′ b e t wo simply c onne cte d ﬁnite rhombic emb e ddings of b-qu ad-gr aphs with the same e dge dir e ctions. Assu me that D and D ′ agr e e on al l b oundary fac es. L et C b e an (emb e dde d) planar cir cle p attern for D and the lab el ling given by the rhombic emb e dding. Then ther e is an (emb e dde d) planar cir cle p attern C ′ for D ′ which agr e es with C for al l b oundary cir cles. Pr o of. Cons ider the monotone com binato rial surfaces Ω D and Ω ′ D . Without loss of gener ality , we can assume that Ω D and Ω ′ D hav e the same b oundary fac e s in Z d . Thus they b oth deﬁne the same br ick Π(Ω D ) = Π(Ω ′ D ) =: Π. Given the cir- cle pattern C , deﬁne the function w on V (Ω D ) by (14). E x tend w to the brick Π such that co ndition (15) holds for all tw o-dimensio nal facets. Conside r w on Ω ′ D and build the corre sp onding pa ttern C ′ , such that the p oints o n the bo undary agree with those of the given cir cle pa ttern C . Equation (15) guar a ntees that all rhombi o f Ω ′ D are mapp ed to closed kites. Due to the combinatorics, the chain of kites is clo sed around each vertex. Since the b oundary kites of C ′ are given by C which is a n immersed circle pattern, at every interior white p oint the a ngles of the kites s um up to 2 π . Thus C ′ is an immerse d circle pattern. F urthermore, C ′ is embedded if C is, b eca us e C ′ is an immersed cir c le pattern a nd C ′ and C agree for all b oundar y kites . 5 C ∞ -con v ergence for quasicrystallic circle pat- terns In order to impr ove the order of c o nv ergenc e in Theo rem 3.1 we study partial deriv atives of the ex tended ra dius function using the integrability of the Hiro ta equation (15) and a Regularity Le mma 5.6. The following constants ar e useful to estimate the p os sible o r ders of partia l deriv atives for a function deﬁned on F κ (Ω D ). Note that F κ (Ω D ) ⊂ Π(Ω D ). Deﬁnition 5.1. Let D b e a rhombic embedding of a ﬁnite simply co nnected b-quad-gr aph with corresp onding co m binatorial surface Ω D in Z d . Let J ⊂ { 1 , . . . , d } contain at least tw o diﬀerent indices. C ∞ -convergence for quasicr yst allic circle p a tterns 23 F or B ≥ 0 deﬁne a c ombinatorial b al l of radius B ab out z ∈ V ( Z d ) us ing the dir ections { e j : j ∈ J } by U J ( z , B ) = { ζ = z + X j ∈ J n j e j : X j ∈ J | n j | ≤ B } . (38) The r adius of the lar gest b al l a bo ut z using these directions which is contained in F κ (Ω D ) is denoted by B J ( z , F κ (Ω D )) = max { B ∈ N : U J ( z , B ) ⊂ F κ (Ω D ) } . Denote by d ( ˆ z , ∂ Ω D ) the c ombinatorial distanc e of ˆ z ∈ V (Ω D ) to the b ound- ary ∂ Ω D , that is the smallest int eger K such that there is a co nnected path with K edges contained in Ω D from ˆ z to a b oundar y vertex of ∂ Ω D . F or further use, we deﬁne the co nstant C J ( F κ (Ω D )) = min  B J ( ˆ z , F κ (Ω D )) + 1 d ( ˆ z , ∂ Ω D ) : ˆ z ∈ V int (Ω D )  > 0 . (39 ) Note as an immediate consequence that for all ˆ z ∈ V int (Ω D ) U J ( ˆ z , ⌈ C J ( F κ (Ω D )) d ( ˆ z , ∂ Ω D ) − 1 ⌉ ) ⊂ F κ (Ω D ) , where ⌈ s ⌉ deno tes the smallest int e ger bigger than s ∈ R . The following theor em is an impr ov ed version o f Theor em 3.1 for a s p eciﬁe d class of q uasicrysta llic cir cle pa tter ns. Theorem 5.2. Un der t he assu mptions of The or em 3.1 and with the same no- tation, let d ∈ N with d ≥ 2 b e a c onstant and assume further that D n is a qu a- sicrystal lic rhombic emb e dding in D with e dge lengths ε n and dimension d n ≤ d . The dir e ctions of the e dges ar e elements of the set {± a ( n ) 1 , . . . , ± a ( n ) d n } ⊂ S 1 such that any two of the ve ctors of { a ( n ) 1 , . . . , a ( n ) d n } ar e line arly indep endent. The p ossible angles ar e uniformly b oun de d, t hat is for al l n ∈ N the sc alar pr o duct is strictly b ounde d away fr om 1 , |h a ( n ) i , a ( n ) j i| ≤ co s( π/ 2 + C ) < 1 , (40) for al l 1 ≤ i < j ≤ d n and s ome c onstant 0 < C < π / 2 . Conse quently, the interse ction angles α n ar e uniformly b oun de d in the sense that for al l n ∈ N and al l fac es f ∈ F ( D n ) ther e holds | α n ( f ) − π/ 2 | ≤ C . (41) L et κ ∈ N , let J 0 ⊂ { 1 , . . . , d } c ontain at le ast two indic es, and let B , C J 0 > 0 b e r e al c onstants. Supp ose that J 0 ⊂ { 1 , . . . , d n } for al l n ∈ N and C J 0 ( F κ (Ω D n )) ≥ C J 0 > 0 . Then we have with the same deﬁnitions of r n , φ n , q n , and p n as in The o- r em 3.1 t hat q n → g ′ and g n → g in C ∞ ( D ) as n → ∞ , t hat is discr ete p artial derivatives of al l or ders of q n and g n c onver ge uniformly on c omp act subsets to their sm o oth c ounterp arts. Simple examples of seq uences of quasicrys tallic circle pa tterns for this theo- rem a r e subgra phs o f the s uitably sc a led inﬁnite regular squar e grid o r hexago- nal circle pa tterns or subgraphs of suitably sca led inﬁnite rho mbic embeddings 24 Ulrike B ¨ ucking constructed in E xamples 4.2 and 4.3 (see Figure 2). Simply connected par ts of these rho mbic embeddings which are large eno ugh satisfy the conditions C J ( F κ (Ω D n )) ≥ C 0 > 0 for all subsets J ⊂ { 1 , . . . , d } , wher e the cons tant C 0 only dep e nds on the construction par ameters. This is a consequence of the simple combinatorics or of the modiﬁed cons truction in Example 4.3. Remark 5.3. Similarly as for Theorem 3.1, the pr o of of Theorem 5.2 shows that we ha ve in fact a priori b ounds in O ( ε n ) on compact subset o f D for the diﬀerence of directional deriv atives of log g ′ (and th us of g ′ and g ) and corres p o nding discrete partial deriv a tives o f log r n ε n + i ( ϕ n − φ n ) (a nd th us q n and g n resp ectively). The constant in the estimation dep ends on the order of the pa rtial deriv atives and on the co mpact subse t. F urthermore, for the ’very r egular’ case considered in Rema r k 3.4 the a prio ri bo unds on the ab ove pa rtial deriv atives hav e order O ( ε 2 n ) on co mpa ct subse t o f D due to the impr ov ed estimatio n (29). Remark 5. 4. Ther e is a n analo gous v er sion of Theo rem 5.2 of C ∞ -conv erg ence for q ua sicrystallic circle pa tter ns with Neumann b o undary conditions. F or the pro of of Theo rem 5.2, we ﬁrst deﬁne the co mparison function w n for the cir cle pa ttern C n according to (14) and extend it to F κ (Ω D n ). This extension is again denoted by w n . The restrictio n o f w n to white vertices is called extende d r adius function and denoted by r n as the original radius function for C n . W e also ex tend | g ′ | to F κ (Ω D n ) by deﬁning | g ′ ( z ) | to b e the v alue | g ′ ( z ) | at the pro jection z o f z ∈ Z d onto the plane o f the rhombic embedding D n . If n is big enough, which will b e assumed in the following, then z ∈ D or z ∈ W \ D and the dis tance o f z to D is b ounded indep endently of n . Denote t n = log( r n /R n ) = lo g ( r n /ε n ) and h n = log | g ′ | as in Section 3. Using the extensions of r n and | g ′ | , these functions are deﬁned on all white vertices of F κ (Ω D n ). F urthermore we hav e the follo wing extension of Lemma 3.3. Lemma 5.5. The estimation h n ( z ) − t n ( z ) = O ( ε n ) . holds for al l white vertic es z ∈ V ( F κ (Ω D n )) . The c onstant in the O -notation may dep end on κ and on the dimension d n of D n . Pr o of. Let z ∈ V ( F (Ω D n )) b e a white vertex. By deﬁnition of F (Ω D n ) ther e is a combinatorial surface Ω ′ ( z ) containing z with the same bo unda ry curve as Ω D n . Lemma 4.5 implies that we c a n deﬁne an embedded c ircle pattern C ′ using the v alues of w n on Ω D n . Now the cla im follows fro m Lemma 3.3. F or z ∈ V ( F κ (Ω D n )) \ V ( F (Ω D n )) observe that equation (15) may be used to extend the estimatio n of h n − t n . Ea ch steps adds a n er r or of order ε n , therefore the ﬁna l cons ta nt dep ends on κ and on d n . Our main a im is to estimate the partial der iv atives of the extended radius function in Z d n . Suc h partial deriv atives ca n generally b e considered in direction of the vectors v = ± e j 1 ± e j 2 for 0 ≤ j 1 , j 2 ≤ d n such that e j 1 and e j 2 are not collinear. Let v 1 , . . . , v 2 d n ( d n − 1) be an enumeration of these vectors and set V n = { v 1 , . . . , v 2 d n ( d n − 1) } . The corr esp onding en umeration of the directions v = ± a ( n ) j 1 ± a ( n ) j 2 in D n for 0 ≤ j 1 , j 2 ≤ d is denoted by v 1 , . . . v 2 d n ( d n − 1) . C ∞ -convergence for quasicr yst allic circle p a tterns 25 F or a ny function h on white vertices z ∈ Z d and/or in z ∈ C deﬁne discr ete p artial derivatives in direction v i or v i by ∂ v i h ( z ) = h ( z + v i ) − h ( z ) ε n | v i | and ∂ v i h ( z ) = h ( z + ε n v i ) − h ( z ) ε n | v i | resp ectively . F urthermore, we call a directio n v i or v i to b e c ontaine d in Ω D n or D n at a vertex ˆ z o r z respectively if there is a tw o -dimensional fac e t of Ω D n incident to ˆ z who se dia gonal incident to ˆ z is parallel to v i , that is { ˆ z + λ v i : λ ∈ [0 , 1] } ⊂ Ω D n . Corresp o nding to these pa r tial deriv atives we use a sc a led version of the Laplacian in (30). F or a function η and an interior v e rtex z with incident vertices z 1 , . . . , z L deﬁne the discr ete L aplacian b y ∆ ε n η ( z ) := ∆ ε n v µ 1 ,..., v µ L η ( z ) := 1 ε 2 n L X j =1 2 f ′ α n ([ z ,z j ]) (0)( η ( z j ) − η ( z )) . (42) Here v µ j = v ([ z , z j ]) = ( z j − z ) /ε n ( j = 1 , . . . , L ) and the no tation ∆ ε n v µ 1 ,..., v µ L emphasizes the dep endence of the Laplacian on the directions v ([ z , z j ]) of the edges [ z , z j ] ∈ E ( G n ). Let z 0 ∈ V int ( G n ) b e an in terio r vertex and let v µ 1 , . . . , v µ L ∈ V n be the directions whic h corr e sp ond to the directions of the edges of G n incident to z 0 . Let z 1 ∈ F κ (Ω D n ) b e a vertex such that z 1 + v µ i ∈ F κ (Ω D n ) for all i = 1 , . . . , L . Our next aim is to study ∆ ε n v µ 1 ,..., v µ L t n ( z 1 ). F or this purp ose we ass ume that we hav e translated to z 1 the facets of Ω D n incident to z 0 , that is we consider the (very small) monotone surface consisting of the tw o -dimensional facets incident to z 1 which cont ain a diag onal { z 1 + λ v µ i : λ ∈ [0 , 1] } for i = 1 , . . . , L . Using the extension of the comparison function w n , the closed chain of these tw o- dimensional facets incident to z 1 is mapp ed to a close d chain of kites. Thus we hav e L X l =1 2 f α µ l ( t n ( z 1 + v µ l ) − t n ( z 1 )) ∈ 2 π N , where α µ l denotes the lab elling of the tw o-dimensional fa cet cont aining z 1 and z 1 + v µ l . By assumption P L l =1 2 f α µ l (0) = 2 π and we know that t n ( z 1 + v µ l ) − t n ( z 1 ) = O ( ε n ) by Lemma 5.5. Since the in tersection angles and thus the maximum num b er o f neig hbors a re unifor mly b ounded, we deduce that L X l =1 2 f α µ l ( t n ( z 1 + v µ l ) − t n ( z 1 )) ! − 2 π = 0 if ε n is small enoug h. Using a T aylor expansion abo ut 0 , we obtain ∆ ε n v µ 1 ,..., v µ L t n ( z 1 ) = 2 ε 2 n − L X l =1 ∞ X m =3 f ( m ) α µ l (0) m ! ( t n ( z 1 + v µ l ) − t n ( z )) m ! = ε n − 2 L X l =1 ∞ X m =3 f ( m ) α µ l (0) m ! | v µ l | m ( ∂ v µ l t n ( z 1 )) m ε m − 3 n ! =: ε n F v µ 1 ,..., v µ L ( ε n , ∂ v µ 1 t n , . . . , ∂ v µ L t n ; z 1 ) , (43) 26 Ulrike B ¨ ucking Note that F v µ 1 ,..., v µ L is a C ∞ -function in the v ar iables ε n , ∂ v µ 1 t n , . . . , ∂ v µ L t n . This fact will b e impo r tant fo r the pro of of L e mma 5.7 b elow. F or further use we intro duce the following notation. Let K be a compact subset of D and let M > 0. Denote by Ω K,M D n the par t of Ω D n with vertices of combinatorial distance bigger than M to the b o undary and whose cor r esp onding vertices z ∈ V ( D n ) lie in K . Let J ⊂ { 1 , . . . , d } con tain at leas t t wo diﬀerent indices. In order to consider pa rtial deriv atives within K in the directions ± e j 1 ± e j 2 , wher e j 1 , j 2 ∈ J and j 1 6 = j 2 , we attach a ball U J ( ˆ z , M ) at ea ch of these po ints: U J ( K, M , Ω D n ) = [ ˆ z ∈ V (Ω K,M D n ) U J ( ˆ z , M ) . (44) Note that if M ≤ B J ( ˆ z ), for example if M ≤ C J ( F κ (Ω D n )) d ( ˆ z , ∂ Ω D n ) − 1 for all ˆ z ∈ V (Ω K,M D n ), then U J ( K, M , Ω D n ) ⊂ F κ (Ω D n ). F urthermore we deﬁne K + d 0 to b e the co mpa ct d 0 -neighborho o d of the compact s et K ⊂ C , that is K + d 0 = { z ∈ C : e ( K, z ) ≤ d 0 } , where e denotes the Euclidean distanc e b etw een a p oint and a co mpact se t or betw een tw o compact subsets of C . The following lemma is imp or tant for o ur ar gumentation, a s it gives an es ti- mation o f a partia l deriv ative using es tima tions of the function and its La placian. Lemma 5.6 (Regular it y Lemma) . L et D b e a qu asicrystal lic rhombic emb e dding with asso ciate d gr aph G and lab el ling α . L et W ⊂ V ( G ) and let u : W → R b e any function. L et M ( u ) = ma x v ∈ W int | ∆ u ( v ) / (4 F ∗ ( v )) | , wher e F ∗ ( v ) = 1 4 X [ z ,v ] ∈ E ( G ) c ([ z , v ]) | z − v | 2 = 1 2 X [ z ,v ] ∈ E ( G ) sin α ([ z , v ]) is the ar e a of the fac e of the dual gr aph G ∗ c orr esp onding to the vertex v ∈ V int ( G ) . The r e ar e c onstants C 5 , C 6 > 0 , indep endent of W and u , such that | u ( x 0 ) − u ( x 1 ) | ρ ≤ C 5 k u k W + ρ 2 C 6 M ( u ) (45) for al l vertic es x 1 ∈ W incident to x 0 , wher e ρ is the Euclide an distanc e of x 0 to the b oun dary W ∂ . A pr o of is given in the app endix, see L e mma A.8. As a dir ect a pplica tion we g et Lemma 5 . 7. L et K ⊂ D b e a c omp act set , n 0 ∈ N and 0 < d 0 < e ( K, ∂ D n ) for al l n ≥ n 0 . L et k ∈ N 0 and let v i 1 , . . . , v i k ∈ T n ≥ n 0 V n b e k not n e c essarily diﬀer- ent dir e ctions. L et J ⊂ { 1 , . . . , d } b e a minimal su bset of indic es such that { v i 1 , . . . , v i k } ⊂ sp an { e j : j ∈ J } . L et B 0 , C 0 ≥ 0 b e some c onst ants. Assume that al l discr ete p artial derivatives u s ing at most k of the dir e ctions v i 1 , . . . , v i k exist on U 0 = U J ( K + d 0 , B 0 , Ω D n ) and ar e b ounde d on U 0 by C 0 ε n for al l n ≥ n 0 . L et n ≥ n 0 and let Ω ˆ D n b e a two-dimensional monotone c ombinatorial su r- fac e. L et ˆ D n b e the c orr esp onding rhombic emb e dding with e dge lengths ε n and such that { ˆ z ∈ V (Ω ˆ D n ) : z ∈ V ( ˆ D n ) ∩ ( K + d 0 ) } ⊂ U 0 . C ∞ -convergence for quasicr yst allic circle p a tterns 27 L et z 0 ∈ V w ( ˆ D n ) ∩ ( K + d 0 / 2) b e a white vertex s u ch that e ( z , ∂ ˆ D n ) ≥ d 0 / 2 . L et v i k +1 ∈ V b e a dir e ction c ontaine d in Ω ˆ D n at ˆ z 0 . Then ther e is a c onstant C 1 , dep en ding on K , D , g , and on the c onst ants d 0 , B 0 , C 0 , κ , C , but not on z 0 , v i k +1 , and n , such that | ∂ v i k +1 ∂ v i k · · · ∂ v i 1 ( t n − h n )( z 0 ) | ≤ C 1 ε n . Pr o of. The pr o of is an applicatio n o f the Regular ity Lemma 5.6 for the function u = ∂ v i k · · · ∂ v i 1 ( t n − h n ) and the part of the rhombic embedding ˆ D n contained in K + d 0 . N o te that the edg e lengths of ˆ D n are ε n and we s upp o s e that e ( z , ∂ ˆ D n ) ≥ d 0 / 2. By assumption we hav e k u k V ( ˆ D n ) ∩ ( K + d 0 ) ≤ C 0 ε n . As h n is a C ∞ -function, this implies that ∂ v i ∂ v i k · · · ∂ v i 1 t n = O (1 ) on U 0 for all p oss ible directions v i ∈ V whenever this partial deriv ative is deﬁned in F κ (Ω ˆ D n ). Let z 1 ∈ V int ( ˆ G n ) ∩ ( K + d 0 ) be an interior white v er tex of ˆ D n and let v µ 1 , . . . , v µ L ∈ V n be the directions corr e s p o nding to the dire c tions of the edges of ˆ G n incident to z 1 . Note that ∆ ε n v µ 1 ,..., v µ L u = ∆ ε n v µ 1 ,..., v µ L ∂ v i k · · · ∂ v i 1 ( t n − h n ) = ∂ v i k · · · ∂ v i 1 ∆ ε n v µ 1 ,..., v µ L ( t n − h n ) . F rom the ab ov e consideratio n in (43) we use tha t ∆ ε n v µ 1 ,..., v µ L t n = ε n F v µ 1 ,..., v µ L , where F v µ 1 ,..., v µ L is a C ∞ -function in the v a r iables ε n , ∂ v µ 1 t n , . . . , ∂ v µ L t n . F rom our assumptions we know that all pa rtial der iv atives ∂ v µ i ∂ v i k · · · ∂ v i 1 t n ( z 1 ) and those containing less than k + 1 of the deriv atives ∂ v µ i , ∂ v i k , . . . , ∂ v i 1 are deﬁned and uniformly b ounded by a constant indep endent of z 1 and ε n . Thus    ∂ v i k · · · ∂ v i 1 ∆ ε n v µ 1 ,..., v µ L t n ( z 1 )    ≤ C v µ 1 ,..., v µ L ε n , where the co ns tant C v µ 1 ,..., v µ L do es not dep end on z 1 and ε n . As h n is a harmonic C ∞ -function, w e obtain by similar reasonings as in the pro of of Lemma 3.3 tha t ∂ v i k · · · ∂ v i 1 ∆ ε n v µ 1 ,..., v µ L h n = O ( ε n ) . As V n is a ﬁnite set, we deduce that ∆ ε n u is uniformly b o unded on V int ( ˆ G n ) ∩ ( K + d 0 ) b y C 2 ε n for some constant C 2 independent of ε n . Now the Regularity Lemma 5.6 g ives the claim. The following corollar y of the pr eceeding lemma constitutes the crucial step in o ur pr o of o f T he o rem 5.2. Lemma 5.8. L et K ⊂ D b e a c omp act set and let 0 < d 1 < e ( K, ∂ D ) . L et n 0 ∈ N b e such that K + d 1 is c over e d by t he rhombi of D n for al l n ≥ n 0 . Then ther e is a c onst ant C 1 = C 1 ( K, C J 0 ( F κ (Ω D n ))) > 0 su ch that for al l z ∈ K + d 1 C 1 ε − 1 n ≤ C J 0 ( F κ (Ω D n )) · d ( z , ∂ D n ) − 1 . F urthermor e, let k ∈ N 0 and let v i 1 , . . . , v i k ∈ V b e k (not ne c essarily diﬀer ent) dir e ctions such that v i l ∈ sp an { e j : j ∈ J 0 } for al l l = 1 , . . . , k . Then ther e ar e c onstants n 0 ≤ n 1 ( k , K ) ∈ N and C ( k , K ) > 0 which may dep end on k , K , d 1 , D , g , κ , C J 0 ( F κ (Ω D n )) , but not on ε n , su ch that for al l n ≥ n 1 ( k , K ) we have k ∂ v i k · · · ∂ v i 1 ( t n − h n ) k U k ≤ C ( k, K ) ε n , (46) wher e U k = U J 0 ( K + 2 − k d 1 , 2 − k C 1 ε − 1 n , Ω D n ) . 28 Ulrike B ¨ ucking Pr o of. The existence of the consta nt C 1 follows from the fact that d ( z , ∂ D n ) /ε n is bo unded fr om below for z ∈ ( K + d 1 ) since the distance e ( K + d 1 , ∂ D n ) > 0 is p ositive and the angles o f the rho mb i ar e uniformly b ounded. The pr o of of es tima tio n (46) uses induction on the num b er o f par tial deriv a- tives k . F or k = 0 the claim has b een shown in Lemma 5 .5. Let k ∈ N 0 and assume that the cla im is true for all ν ≤ k . Let v i k +1 ∈ V be a direction with v i k +1 ∈ {± e j 1 ± e j 2 } ⊂ span { e j : j ∈ J 0 } . Using the induction hypotheses , we ca n apply Lemma 5.7 for U 0 = U k , d 0 = 2 − k d 1 , Ω ˆ D n = U { j 1 ,j 2 } ( K + 2 − k d 1 , 2 − k C 1 ε − 1 n , Ω D n ) ⊂ U k and the corresp o nding r hom- bic embedding ˆ D n obtained by pro jection, and z 0 ∈ ˆ D n ∩ ( K + 2 − k − 1 d 1 ). This completes the induction step and the pro o f. Pr o of of The or em 5.2. Identify C with R 2 in the standar d way a nd ﬁx t wo or- thogonal unit vectors e 1 , e 2 . Deﬁne discrete partial deriv atives ∂ e 1 , ∂ e 2 in thes e directions using the discrete par tial deriv atives in t wo o rthogona l directions v i 1 = a ( n ) j 1 + a ( n ) j 2 and v i 2 = a ( n ) j 1 − a ( n ) j 2 for j 1 , j 2 ∈ J 0 . This deﬁnition dep ends on the choice of a ( n ) j 1 , a ( n ) j 2 , which may b e diﬀerent for each n , but this do es not aﬀect the pro of. As the p oss ible intersection a ngles ar e b o unded and a s h n is a C ∞ -function, w e deduce that k ∂ e j k · · · ∂ e j 1 h n − ∂ j k · · · ∂ j 1 h n k K ≤ C 1 ( k , K ) ε n on every compa ct set K for j k , . . . , j 1 ∈ { 1 , 2 } . Here ∂ 1 , ∂ 2 denote the sta nda rd partial deriv atives a sso ciated to e 1 , e 2 for s mo oth functions and C 1 ( k , K ) is a constant which dep ends only on K , k , and g . Lemma 5.8 implies that k ∂ e j k · · · ∂ e j 1 t n − ∂ e j k · · · ∂ e j 1 h n k U J 0 ( K +2 − k d 1 , 2 − k C 1 ε − 1 n , Ω D n ) ≤ C 2 ( k , K ) ε n if n is big enough. Using a version of Lemma 3.9 with e r ror of order O ( ε n ), we deduce that t n + i ( ϕ n − φ n ) conv erg es to log g ′ in C ∞ ( D ). Now the conv erg ence of q n and g n follows by s imila r a rguments a s in the pro of of Theo rem 3.1. A App endix: Prop erties of discrete Green’s func- tion and regulari t y of solutions of discrete el- liptic equations In order to prove the Regularity Lemma 5.6 (see Lemma A.8), we present some results in discr ete po tential theory on quas icrystallic rhombic em b eddings which are derived from a suitable asymptotic expansio n of a discr ete Green’s function. Throughout this app endix, we assume that D is a (p ossibly inﬁnite) simply connected q ua sicrystallic rho mbic embedding of a b-quad-graph with edge di- rections A = {± a 1 , . . . , ± a d } . Also, the edge lengths of D are supp os e d to b e normalized to one. Let G be the asso ciated gra ph built from white vertices. Fix some interior vertex x 0 ∈ V int ( G ). F o llowing Keny on [Ken02] and Bob enko, Mercat, and Sur is [BMS05], w e deﬁne the discr ete Gr e en ’s function G ( x 0 , · ) : V ( G ) → R by G ( x 0 , x ) = − 1 4 π 2 i Z Γ log( λ ) 2 λ e ( x ; λ ) dλ (47) Appendi x: Regularity of solutions of discrete elliptic equa tions 29 for all x ∈ V ( G ). Here e ( x ; z ) = Q d k =1  z + a k z − a k  n k is the discr ete ex p onential function , where n = ( n 1 , . . . , n d ) = ˆ x − ˆ x 0 ∈ Z d and ˆ x , ˆ x 0 ∈ V (Ω D ) cor r esp ond to x, x 0 ∈ V ( D ) resp ectively . The in teg ration path Γ is a collectio n of 2 d small loo ps , eac h one r unning co un terclo ckwise around one of the p oints ± a k for k = 1 , . . . , d . The branch of log( λ ) depends on x and is c hosen a s follows. Without loss of generality , we assume that the circula r or der of the p oints of A o n the p ositively oriented unit circle S 1 is a 1 , . . . , a d , − a 1 , . . . , − a d . Set a k + d = − a k for k = 1 , . . . , d and deﬁne a m for all m ∈ Z by 2 d -p erio dicity . T o each a m = e iθ m ∈ S 1 we assig n a ce r tain v alue o f the a rgument θ m ∈ R : cho ose θ 1 arbitrar ily and then use the rule θ m +1 − θ m ∈ (0 , π ) fo r a ll m ∈ Z . Clearly we then hav e θ m + d = θ m + π . The p oints a m supplied with the arguments θ m can be considered as belo ng ing to the Riemann sur face of the log arithmic function (i.e. a branched cov ering of the complex λ -plane). Since Ω D is a mono- tone sur face, ther e is a n m ∈ { 1 , . . . , 2 d } and a directed path from x 0 to x in D suc h tha t the dir ected edg es of this path are contained in { a m , . . . , a m + d − 1 } , see [BMS0 5, Lemma 18]. Now, the branch of log( λ ) in (47) is chosen such that log( a l ) ∈ [ i θ m , iθ m + d − 1 ] , l = m, . . . , m + d − 1 . Remem ber Deﬁnition 3.5 of the discrete Lapla cian and the representation of its w eig hts c ([ z 1 , z 2 ]) = 2 f ′ α ([ z 1 ,z 2 ]) (0) in (4). Then there holds Lemma A. 1 ([K e n02, Theor ems 7 .1 and 7.3 ]) . The discr ete Gr e en ’s function G ( x 0 , · ) deﬁne d in e quation (47) has the fol lowing pr op erties. (i) ∆ G ( x 0 , v ) = − δ x 0 ( v ) , wher e the L aplacian is t aken with r esp e ct to the se c ond variable. (ii) G ( x 0 , x 0 ) = 0 . (iii) G ( x 0 , v ) = O (log ( | v − x 0 | )) . Note that G ( x 0 , · ) may also be deﬁned by these three conditio ns . A.1 Asymptotics for discret e Green’s function Keny on der ived in [Ken02] a n asymptotic developmen t for the discr e te Green’s function using standar d metho ds of c o mplex analy sis. His r esult ca n b e s lightly strengthened to an er ror of order O (1 / | v − x 0 | 2 ). Note, that there is the sum- mand − log 2 / (2 π ) missing in Keny o n’s formula (but not in his pro of ). Theorem A. 2 (cf. [Ken02, Theorem 7 .3]) . F or v ∈ V ( G ) ther e holds G ( x 0 , v ) = − 1 2 π log(2 | v − x 0 | ) − γ Euler 2 π + O  1 | v − x 0 | 2  . (48) Her e γ Euler denotes the Euler γ constan t . 30 Ulrike B ¨ ucking Pr o of. Cons ider a directed path x 0 = w 0 , . . . , w k = v in D fro m x 0 to v such that the directed edges of this path are contained in { a m , . . . , a m + d − 1 } for some 1 ≤ m ≤ 2 d as ab ov e. Note that k is even s inc e x 0 and v are b oth white vertices of D . The integration path Γ in (47) can b e deformed into a connected contour lying on a single lea f of the Riemann surface of the loga rithm, in particular to a simple clos e d curv e Γ 1 which surro unds the set { a m , . . . , a m + d − 1 } in a counterclockwise sense a nd has the or igin and a ra y R = { s e i ˜ θ : s > 0 } in its exterior. W e also as s ume that Γ 1 is contained in the secto r { z = r e iϕ : r > 0 , ϕ ∈ [ ˜ θ + π 2 + η , ˜ θ + 3 π 2 − η ] } for some η > 0 indep endent of v , x 0 , and m . This is po ssible due to the fact that θ m + d − 1 − θ m < π − δ for some δ > 0 indepe ndent of v , x 0 , and m . Let N = | v − x 0 | . T ake 0 <  1 ≪ 1 / N 3 and  2 ≫ N 3 , but not exp onentially smaller than 1 / N or bigge r than N resp ectively . The curve Γ 1 is a gain homo - topic to a curve Γ 2 which runs counterclockwise around the circle of radius  2 ab out the o rigin from the angle ˜ θ to ˜ θ + 2 π , then a lo ng the ray R fro m  2 e i ˜ θ to  1 e i ˜ θ , then clo ckwise around the circle of r adius  1 ab out the orig in fr om the angle ˜ θ + 2 π to ˜ θ , and ﬁnally back a long the ray R from  1 e i ˜ θ to  2 e i ˜ θ . Without loss of genera lity , we ass ume that R is the ne g ative real ax is. Keny on showed in [Ken02] that the in teg rals along the circles of r adius  1 and  2 give ( − 1) k log  1 4 π (1 + O ( N  1 )) − log  2 4 π (1 + O ( N / 2 )) . The diﬀerence b etw een the v alue of log z ab ove and b elow the negative real axis is 2 π i . Thus the integrals along the negative rea l axis ca n b e combined into − 1 4 π Z −  1 −  2 1 z k − 1 Y j =0 z + b j z − b j dz , where b j = w j +1 − w j ∈ A is the directed edg e from w j to w j +1 and k = O ( N ) is the num b er o f edges of the path. This int e gral can b e split into thr ee parts: from −  2 to − √ N , from − √ N to − 1 / √ N , and from − 1 / √ N to −  1 . The in teg r al is neglectible for the intermediate r ange b ecaus e     t + e iβ t − e iβ     ≤ e 2 t cos β / ( t − 1) 2 for neg ative t < 0 and due to our assumptions. F or s mall | t | we have k − 1 Y j =0 t + b j t − b j = k − 1 Y j =0 t ¯ b j + 1 t ¯ b j − 1 = ( − 1) k e 2 P k − 1 j =0 ¯ b j t (1 + O ( k t 3 )) , using the Neumann series a nd a T a ylo r expansion. Thus the integral nea r the origin is − ( − 1) k 4 π Z −  1 − 1 / √ N e 2( ¯ v − ¯ x 0 ) t t dt + Z −  1 − 1 / √ N O ( k t 3 ) e 2( ¯ v − ¯ x 0 ) t t dt ! . Appendi x: Regularity of solutions of discrete elliptic equa tions 31 Applying similar r easoning s and estimatio ns as in K eny on’s pr o of, we obtain − ( − 1) k 4 π (log(2  1 ( ¯ v − ¯ x 0 )) + γ Euler ) + O  1 N 2  . Here γ Euler denotes the Euler γ constan t. F or la rge | t | the estimations are v ery similar . Since k − 1 Y j =0 t + b j t − b j = k − 1 Y j =0 1 + b j t − 1 1 − b j t − 1 = e 2 P k − 1 j =0 b j t − 1 (1 + O ( k t − 3 )) , we get − ( − 1) k 4 π Z − √ N −  2 e 2( v − x 0 ) t − 1 t dt + Z − √ N −  2 O ( k t − 3 ) e 2( v − x 0 ) t − 1 t dt ! = − 1 4 π  − log   2 2( v − x 0 )  + γ Euler  + O  1 N 2  . Since k is even, the sum of all the ab ov e integral parts is therefore given by the r ight hand side of (48). F or a bounded domain w e also deﬁne a discrete Gr een’s function with v anish- ing b o undary v alues. L e t W ⊂ V int ( G ) b e a ﬁnite subset of vertices. Denote by W ∂ ⊂ W the set of b o unda ry vertices which are inciden t to at least one vertex in V ( G ) \ W . Set W int = W \ W ∂ the interior vertices o f W . Let x 0 ∈ W int be an interior vertex. The di scr et e Gr e en ’s funct ion G W ( x 0 , · ) is uniq ue ly deﬁned by the following pro pe r ties. (i) ∆ G W ( x 0 , v ) = − δ x 0 ( v ) for all v ∈ W int , where the Lapla cian is ta ken with resp ect to the second v ariable. (ii) G W ( x 0 , v ) = 0 for all v ∈ W ∂ . In the following, we c ho ose W to be a sp ecial disk-like set. Let x 0 ∈ V ( G ) be a vertex a nd let ρ > 2. Denote the c losed disk with cen ter x 0 and radius ρ by B ρ ( x 0 ) ⊂ C . Supp ose that this disk is entirely cov ered b y the rhombic embedding D . Denote by V ( x 0 , ρ ) ⊂ V ( G ) the set o f white vertices lying within B ρ ( x 0 ). F or x 1 ∈ V int ( x 0 , ρ ) we denote G x 0 ,ρ ( x 1 , · ) = G V ( x 0 ,ρ ) ( x 1 , · ) . The asymptotics of the discrete Green’s function G from The o rem A.2 ca n be used to derive the following estimations fo r G x 0 ,ρ . Prop ositi on A. 3 . Ther e is a c onst ant C 1 , indep en dent of ρ and x 0 , such that |G x 0 ,ρ ( x 0 , v ) | ≤ C 1 /ρ for al l vertic es v ∈ V int ( x 0 , ρ ) which ar e incident to a b oun dary vertex. F urthermor e, ther e is a c onstant C 2 , indep endent of ρ and x 0 , such that for al l interior vertic es x 1 ∈ V int ( x 0 , ρ ) incident to x 0 and al l v ∈ V ( x 0 , ρ ) ther e holds |G x 0 ,ρ ( x 0 , v ) − G x 0 ,ρ ( x 1 , v ) | ≤ C 2 / ( | v − x 0 | + 1) . 32 Ulrike B ¨ ucking Pr o of. Cons ider the function h ρ ( x 0 , · ) : V ( x 0 , ρ ) → R deﬁned b y h ρ ( x 0 , v ) = G x 0 ,ρ ( x 0 , v ) − G ( x 0 , v ) − 1 2 π (log(2 ρ ) + γ Euler ) . Then h ρ ( x 0 , · ) is harmo nic on V int ( x 0 , ρ ). F or b oundar y vertices v ∈ V ∂ ( x 0 , ρ ) Theorem A.2 implies that h ρ ( x 0 , v ) = O (1 /ρ ). The Maximum Pr inciple 3.6 yields | h ρ ( x 0 , v ) | ≤ C /ρ fo r all v ∈ V ( x 0 , ρ ) and so me constant C indep endent of ρ and v . This shows the ﬁrst estima tion. T o prov e the seco nd claim, we also consider the ha rmonic function h ρ ( x 1 , v ) = G x 0 ,ρ ( x 1 , v ) − G ( x 1 , v ) − 1 2 π (log(2 ρ ) + γ Euler ) for a ﬁxed in ter io r vertex x 1 ∈ V int ( x 0 , ρ ) incident to x 0 . By s imilar rea sonings as for h ρ ( x 0 , · ), we deduce tha t | h ρ ( x 1 , v ) | ≤ ˜ C /ρ fo r all v ∈ V ( x 0 , ρ ) and some consta nt ˜ C indep endent of ρ and v . Theo r em A.2 implies the des ir ed estimation. A.2 Regularit y of discrete solutions of elliptic equations In the following, we generalize and ada pt so me results of discrete p otential theory for discr ete harmonic functions obtained by Duﬃn in [Duf53, see in particular Lemma 1 and Theore m 3– 5]. The pro o fs are very s imila r or use idea s of the co rresp o nding pro ofs in [Duf53]. Our aim is to obta in the Reg ularity Lemma A.8. Lemma A.4 (Green’s Identit y) . L et W ⊂ V ( G ) b e a ﬁ nite subset of vertic es. L et u, v : W → R b e t wo functions. Then X x ∈ W int ( v ( x )∆ u ( x ) − u ( x )∆ v ( x )) = X [ p,q ] ∈ E ∂ ( W ) c ([ p, q ])( v ( p ) u ( q ) − u ( p ) v ( q )) , (49) wher e E ∂ ( W ) = { [ p , q ] ∈ E ( W ) : p ∈ W int , q ∈ W ∂ } . Corollary A.5 (Repres ent ation of harmonic functions) . L et u b e a r e al value d harmonic function deﬁne d on V ( x 0 , ρ ) . Then u ( x 0 ) = X q ∈ V ∂ ( x 0 ,ρ ) c ( q ) u ( q ) , wher e c ( q ) = X p ∈ V int ( x 0 ,ρ ) and [ p,q ] ∈ E ( G ) c ([ p, q ]) G x 0 ,ρ ( x 0 , p ) = O (1 /ρ ) . (50) The estimation in (50) is a consequence of Pro p osition A.3 and of the b ound- edness o f the weight s c ( e ). Theorem A.6. L et u : V ( x 0 , ρ ) → R b e a n on-ne gative harmonic function. Ther e is a c onstant C 3 indep endent of ρ and u such that       u ( x 0 ) − 1 π ρ 2 X v ∈ V int ( x 0 ,ρ ) F ∗ ( v ) u ( v )       ≤ C 3 u ( x 0 ) ρ , (51) wher e F ∗ ( v ) = 1 4 P [ z ,v ] ∈ E ( G ) c ([ z , v ]) | z − v | 2 is t he ar e a of the fac e of the dual gr aph G ∗ c orr esp onding to the vertex v ∈ V int ( G ) . Appendi x: Regularity of solutions of discrete elliptic equa tions 33 Pr o of. Consider the function p ( z ) = G ( x 0 , z ) + 1 2 π (log(2 ρ ) + γ Euler ) + | z − x 0 | 2 − ρ 2 4 π ρ 2 . Let z ∈ V int ( G ) b e an interior vertex. Consider the chain o f faces f 1 , . . . , f m of D which are incident to z in coun terclo ckwise order. The enumeration of the faces f j and of the blac k vertices v 1 , . . . , v m and the white vertices z 1 , . . . , z m incident to these faces can b e chosen such that f j is incident to v j − 1 , v j , and z j for j = 1 , . . . , m , where v 0 = v m . F urthermore , using this enumeration we hav e z j − z | z j − z | i = v j − v j − 1 | v j − v j − 1 | ⇐ ⇒ | v j − v j − 1 | | z j − z | ( z j − z ) = − i ( v j − v j − 1 ); see Figure 4 with z − = z , z + = z j , v − = v j − 1 , v + = v j . As | z j − x 0 | 2 − | z − x 0 | 2 = − 2Re(( z − x 0 )( z j − z )) + | z j − z | 2 we deduce by very similar calculations as in the pr o of o f L emma 3.3 that ∆ p ( z ) = ∆ G ( x 0 , v ) + 1 4 π ρ 2 m X j =1 c ([ z , z j ]) | {z } =2 f ′ α ([ z ,z j ]) (0) ( | z j − x 0 | 2 − | z − x 0 | 2 ) = − δ x 0 ( z ) + F ∗ ( z ) / ( π ρ 2 ) . Let v b e incident to a vertex of V ∂ ( x 0 , ρ ). Theorem A.2 implies that p ( v ) = − 1 2 π log | v − x 0 | ρ + | v − x 0 | 2 − ρ 2 4 π ρ 2 + O  1 | v − x 0 | 2  = O (1 /ρ 2 ) . Thu s there is a constant B 1 , indep endent of ρ and v , such that p 1 ( v ) := p ( v ) + B 1 /ρ 2 ≥ 0 and | p 1 ( v ) | ≤ 2 B 1 /ρ 2 for all v er tices v ∈ V int ( x 0 , ρ ) inciden t to a vertex of V ∂ ( x 0 , ρ ). Applying Green’s Identit y A.4 to p 1 and the non-neg a tive harmonic function u , we o bta in u ( x 0 ) − 1 π ρ 2 X v ∈ V int ( x 0 ,ρ ) F ∗ ( v ) u ( v ) = X x ∈ V int ( x 0 ,ρ ) ( p 1 ( x )∆ u ( x ) − u ( x )∆ p 1 ( x )) = X [ z ,q ] ∈ E ρ c ([ z , q ])( p 1 ( z ) u ( q ) − u ( z ) p 1 ( q ) | {z } ≥ 0 ) , ≤ 2 B 1 ρ 2 4 π X q ∈ V ∂ ( x 0 ,ρ ) u ( q ) ≤ 8 π B 1 B 2 ρ u ( x 0 ) , Here E ρ = { [ x, y ] ∈ E ( G ) : x ∈ V int ( x 0 , ρ ) , y ∈ V ∂ ( x 0 , ρ ) } and w e hav e us e d the e s timation X [ x,y ] ∈ E ( G ) c ([ x, y ]) ≤ X [ x,y ] ∈ E ( G ) c ([ x, y ]) | x − y | 2 = 4 F ∗ ( x ) < 4 π for all ﬁxe d vertices x ∈ V int ( G ). F ur ther more P y ∈ V ∂ ( x 0 ,ρ ) u ( y ) ≤ B 2 ρu ( x 0 ) for some co nstant B 2 > 0 as a consequence of Coro llary A.5. F or the reverse inequality , note that there is also a constant B 3 independent of ρ and v such that p 1 ( v ) := p ( v ) − B 3 /ρ 2 ≤ 0 and | p 1 ( v ) | ≤ 2 B 3 /ρ 2 for all vertices v ∈ V int ( x 0 , ρ ) inciden t to a vertex in V ∂ ( x 0 , ρ ). Combining b oth estimation prov es the claim. 34 Ulrike B ¨ ucking Theorem A.6 can be in terpre ted as an a nalog to the Theore m of Ga uss in po tent ial theory . F urthermor e, w e can deduce a discrete v ers ion o f H¨ older ’s Inequality for no n-negative ha rmonic function. Theorem A.7 (H¨ older’s Inequa lity) . L et u : V ( x 0 , ρ ) → R b e a non-ne gative harmonic function. Ther e is a c ons t ant C 4 , indep endent of ρ and u , such that | u ( x 0 ) − u ( x 1 ) | ≤ C 4 u ( x 0 ) /ρ (52) for al l vert ic es x 1 ∈ V ( x 0 , ρ ) incident to x 0 . As a cor ollary of H¨ olde r ’s Inequality and o f Prop os ition A.3 we obtain the following r esult on the regula r ity of discrete so lutions to elliptic equatio ns . Lemma A.8 (Regula rity Lemma) . L et W ⊂ V ( G ) and let u : W → R b e any function. Set M ( u ) = ma x v ∈ W int | ∆ u ( v ) / (4 F ∗ ( v )) | , wher e F ∗ ( v ) is the ar e a of the fac e dual to v as in The or em A.6. Deﬁne k η k W := max {| η ( z ) | : z ∈ W } . Ther e ar e c onstants C 5 , C 6 > 0 , indep endent of W and u , such that | u ( x 0 ) − u ( x 1 ) | ρ ≤ C 5 k u k W + ρ 2 C 6 M ( u ) (53) for al l vertic es x 1 ∈ W incident t o x 0 ∈ W int , wher e ρ is the Euclide an distanc e of x 0 to the b oun dary W ∂ . Pr o of. Let x 1 ∈ W b e a ﬁx ed vertex incident to x 0 . First we supp ose that ρ ≥ 4. Consider the auxilia ry function f ( z ) = M ( u ) | z − x 0 | 2 . Since | x 1 − x 0 | < 2 , we obviously hav e | f ( x 0 ) − f ( x 1 ) | = M ( u ) | x 1 − x 0 | 2 ≤ 4 M ( u ) . Let h : V ( x 0 , ρ ) → R b e the unique ha r monic function with b oundary v alues h ( v ) = u ( v ) + f ( v ) fo r v ∈ V ∂ ( x 0 , ρ ). H¨ older’s Inequality (52) and the Maximum Principle 3.6 imply that | h ( x 0 ) − h ( x 1 ) | ρ ≤ B 1 k h k V ( x 0 ,ρ ) ≤ B 1 ( k u k W + M ( u ) ρ 2 ) for s o me co ns tant B 1 independent o f h , ρ , x 0 , x 1 . Next c onsider s = u + f − h on V ( x 0 , ρ ). Then ( ∆ s = ∆ u + 4 F ∗ M ( u ) ≥ 0 on V int ( x 0 , ρ ) , s ( v ) = 0 for v ∈ V ∂ ( x 0 , ρ ) . The Maximum Pr inciple 3 .6 implies s ≤ 0 . Gree n’s Identit y A.4 gives s ( x 0 ) + X v ∈ V int ( x 0 ,ρ ) G x 0 ,ρ ( x 0 , v )∆ s ( v ) = X v ∈ V int ( x 0 ,ρ ) ( G x 0 ,ρ ( x 0 , v )∆ s ( v ) − s ( v )∆ G x 0 ,ρ ( x 0 , v )) = X [ p,q ] ∈ E ρ c ([ p, q ])( G x 0 ,ρ ( x 0 , p ) s ( q ) − s ( p ) G x 0 ,ρ ( x 0 , q )) = 0 , REFERENCES 35 where E ρ = { [ p, q ] ∈ E ( G ) : p ∈ V int ( x 0 , ρ ) , q ∈ V ∂ ( x 0 , ρ ) } . Analog ously , s ( x 1 ) + X v ∈ V int ( x 0 ,ρ ) G x 0 ,ρ ( x 1 , v )∆ s ( v ) = 0 . Using the es timation ∆ s ( v ) ≤ 8 F ∗ ( v ) M ( u ) we deduce that | s ( x 0 ) − s ( x 1 ) | ≤ X v ∈ V int ( x 0 ,ρ ) |G x 0 ,ρ ( x 0 , v ) − G x 0 ,ρ ( x 1 , v ) | 8 F ∗ ( v ) M ( u ) . Now Prop o sition A.3 implies that | s ( x 0 ) − s ( x 1 ) | ≤ 8 B 2 M ( u ) X v ∈ V int ( x 0 ,ρ ) F ∗ ( v ) | v − x 0 | + 1 ≤ 8 B 2 M ( u ) B 3 ρ, where B 2 and B 3 are cons ta nt s indep endent of s , ρ , x 0 , x 1 . Combining the a bove estimatio ns for f , h , and s , we ﬁnally obtain | u ( x 0 ) − u ( x 1 ) | ρ ≤ | s ( x 0 ) − s ( x 1 ) − ( f ( x 0 ) − f ( x 1 )) + h ( x 0 ) − h ( x 1 ) | ρ ≤ B 1 k u k W + ρ 2 (4 + B 1 + 8 B 2 B 3 ) M ( u ) . This implies the claim for ρ ≥ 4. F or ρ < 4 inequality (53) can b e deduced from − 4 F ∗ ( x 0 ) M ( u ) ≤ ∆ u ( x 0 ) = X [ x 0 ,v ] ∈ E ( G ) c ([ x 0 , v ])( u ( v ) − u ( x 0 )) ≤ 4 F ∗ ( x 0 ) M ( u ) . using F ∗ ( x 0 ) ≤ π , P [ x 0 ,v ] ∈ E ( G ) c ([ x 0 , v ]) ≤ 4 π , a nd the unifor m b oundedness of the weights c ( e ). References [AB00] S. 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Approximation of conformal mappings by circle patterns

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