A Concentration of Measure Phenomenon in Lattice Yang-Mills

A Concen tration of Measure Phenomenon in Lattice Y ang-Mills T. Tlas Abstract W e demonstrate that the pushforw ard of the product of Haar measures b y the lattice Y ang-Mills action concen trates as a Gaussian. It is also sk etched ho w using this fact, one can recov er the strong-coupling expan- sion. The aim of this note is to explore the concentration of measure phenomenon [1] in the con text of lattice Y ang-Mills theory . W e shall see below that while this phenomenon do es indeed manifest in this setting, it do es not allow us to go b ey ond what is already kno wn by using other metho ds. This is due to the fact that, in a certain sense whic h will become clear b elow, measure concentration and action minimization work against eac h other. Nevertheless, the theorem demonstrated b elo w, even though it do es not lead to new results in the curren t setting, is sufficien tly in teresting and instructive to b e presented as the same metho ds can b e (and in fact are) more successful in other situations. Let us b egin by describing our setup. W e shall deal with the Euclidean, lattice Y ang-Mills theory in D dimensions with p erio dic b oundary conditions. F or concreteness, we w ork with the U ( N ) group, but essentially ev erything b elo w can be trivially adapted to the other groups of ph ysical in terest. The lattice sites are labelled by x . The positive edge directions are labelled by greek letters. The notation x + µ stands for the lattice site whic h is displaced from x one step in the direction defined b y µ . The num ber of lattice sites is denoted by K . W e attac h to ev ery edge at each site an element U x,µ ∈ U ( N ), and are interested in p erforming the following integral Z = ˆ dU e − β S ( U ) = ˆ Y x,µ dU x,µ exp " 2 N 2 λ X { µ,ν } 1 N ℜ  T r  U † x,ν U † x + ν,µ U x + µ,ν U x,µ   # , where the sum goes ov er all (unordered) pairs of µ and ν suc h that µ  = ν , or equiv alently , o ver all plaquettes of the lattice. The dU ’s stand for the Haar measures and λ is the ’t Ho oft coupling. W e w ant to find the asymptotic of the expression ab o ve in the limit N → ∞ while holding λ fixed. W e shall do so b y pushing forw ard the pro duct of Haar measures to R via the function t ( U x,µ ) = 1 K X { µ,ν } 1 N ℜ  T r  U † x,ν U † x + ν,µ U x + µ,ν U x,µ   . (1) W e thus get that Z = ˆ e 2 N 2 λ K t dρ N [ t ] , (2) where ρ N is the pushforward measure under the function ab ov e. What can we sa y ab out this measure? The key fact is furnished by the following Theorem. As N → ∞ the r esc ale d me asur es ρ N ( N t ) c onver ge we akly to the me asur e q 2 K π D ( D − 1) e − 2 K D ( D − 1) t 2 dt . Pr o of. Let m l stand for the l -th momen t of Gaussian ab ov e. In other w ords, let m l =    0 if l is o dd ( l − 1)!!  4 K D ( D − 1)  l 2 if l is even . It will b e sho wn below that the l -th momen t of ρ N ( t ) is equal to m l N l + O ( 1 N l +1 ), whic h of course implies that the l -th momen t of ρ N ( N t ) is m l + O ( 1 N ) → m l as N → ∞ . Therefore, com bining the method of moments together with the fact that the moment problem for a Gaussian is determined, the theorem will b e prov en. W e now adapt the graphical representation common in the literature [4] to our goals. A line segment decorated by an arrow will stand for U ij and an adjacen t line segmen t with the arrow p oin ting in the other direction will denote U † i ′ j ′ . Since all the expressions b elo w will b e symmetric b et ween U and U † , we w on’t need to specify which is whic h. An undecorated line segmen t will stand for a Kronec ker δ ij . A ttaching line segmen ts to eac h other corresponds to m ultiplying the corresp onding matrices in the order dictated by thinking of them as being parallel transp orts. See Figure 1 for a summary of this notation. Note that we do not require all these segmen ts to b e straight or horizon tal, as the only fact that matters is how they are connected to eac h other. U ij or U † ij U ij U † i ′ j ′ δ ij U j k U ′ ij P j U ′ ij U j k Figure 1: Summary of a gr aphic al notation use d b elow. Note the or der of the pr o duct in the sum in the rightmost expr ession. It is the one c onsistent with tr e ating U and U ′ as b eing p ar al lel tr ansp orts (acting c onventional ly to the right). W e shall deal with integrals of polynomials in the matrix en tries (as w ell as their conjugates) with resp ect to the Haar measure. Such integrals are w ell-studied with their ev aluations b eing kno wn explicitly for any N (see [2] for a lov ely o verview). How ever, w e w on’t hav e an occasion below to use an ything beyond the low est asymptotic order obtained in [3]. The formula w e need is ˆ dU U i 1 j 1 . . . U i n j n U ∗ i ′ 1 j ′ 1 . . . U ∗ i ′ m j ′ m = δ nm N n X σ δ i 1 i ′ σ (1) δ j 1 j ′ σ (1) . . . δ i n i ′ σ ( n ) δ j n j ′ σ ( n ) + O ( 1 N n +1 ) , (3) where the sum go es o ver all p ermutations of n elements. A graphical represen- tation of the formula abov e for the cases n = 2 and n = 4 is shown in Figure 2. ´ dU = 1 N , ´ dU = 1 N 2 ( + ) + O ( 1 N 3 ) Figure 2: A pictorial r epr esentation of (3) in the c ases n = 2 and n = 4 . Gr aphic al ly, inte gr ation over a bund le of e dges amounts to cutting them into half-e dges, and then linking or c oupling the half e dges in p airs on one side of the cut with e ach other, with matching c oupling of the half-e dges on the other side of the cut as wel l. Incidental ly, note that in the first c ase, the dominant asymptotic happ ens to b e the exact answer. Ther e ar e no further c orr e ctions. W e are no w ready to compute the momen ts of ρ N . F rom the definition of the ρ N w e hav e that ˆ t l dρ N ( t ) = ˆ t l ( U x,µ ) Y x,µ dU x,µ (4) F rom (1), it is obvious that the expression ab o ve w ould be a sum of integrals of pro ducts of l plaquettes. W e thus concentrate on computing one such term. F rom (3), it is clear that the result is nonv anishing if ev ery edge app ears an ev en num b er of times, half of them carrying U and the other half carrying U † . Let us now find the leading asymptotic of eac h such term. First, from (1), note that each plaquette comes with a factor of 1 N . Thus, w e ha ve an o verall factor of 1 N l . Second, from (3), w e hav e that the in tegral con- tributes a further factor of 1 N for each pair of edges in tegrated o ver. Since there are 4 l edges in total, we get a further factor of 1 N 2 l . Finally , again from (3), we can see that integration is going to pro duce pro ducts of delta functions. Since there are no free indices in the expression in tegrated, we will simply hav e a collection of traces of iden tity . Each of them is going to contribute a factor of N . It remains to count how man y suc h traces will b e pro duced. T o this end, pick a pair of edges whic h ha ve been cut with the half-edges coupled after in tegration, and follo w one of the t wo pairs of the coupled half-edges. After reac hing a corner, either these tw o edges will b e coupled again, thus pro ducing a loop, or they will coupled to differen t half-edges. Note that in the former case, w e get a lo op out of 4 half-edges, while in the latter, we will use up more than 4 half-edges to form a lo op. See Figure 3 for a pictorial clarification. ´ dU . . . + . . . a) b) c) Figure 3: A pictorial r epr esentation of the ar gument given in the main text. We p erform the inte gr al over the gr oup elements living on the e dges. Only one of these inte gr als is explicitly written out with the r est b eing r epr esente d by the . . . under the inte gr al sign. As r epr esente d in Figur e 2, after p erforming the inte gr al over the gr oup, the e dges c arrying that gr oup element and its inverse ar e cut into half-e dges with these halves linke d up in p airs in identic al ways on the two sides of the cut. In the figur e, we fo cus on two e dges r epr esente d by solid lines with arr ow whose half-e dges wil l b e c ouple d. We fo cus on only one c ouple d p air, denote d by a solid line. The dashe d one stands for the other one. The dotte d lines stand for the other c opies of the same e dge c arrying U and U † which, after inte gr ation, ar e of c ourse c ouple d among themselves as wel l, which is indic ate d by the br e ak in the lines. The + . . . stands for al l the other terms in the sum in (3) which c orr esp ond to a differ ent c oupling of the half-e dges. Also, note that sinc e our c onc ern her e is the factors of N obtaine d by fol lowing the c oupling of e dges, we haven ’t indic ate d either the factors of 1 N c oming fr om the definition of t ( U ) , or those on the right hand side of (3). Now, fol lowing the two single d out c ouple d half-e dges, we wil l have one of thr e e c ases in the lower half of the figur e. In c ase (a), they me et at the c orner with two c ouple d half-e dges forming a lo op. Note that this lo op is c onstructe d out of four half-e dges. Also note that the two plaquettes fr om which these four half-e dges c ame must actual ly b e the same plaquette, as two of their e dges c oincide. In c ases (b) and (c), the original two half-e dges ar e not c ouple d b ack imme diately forming a lo op, but inste ad, ar e c ouple d to other half e dges. The differ enc e b etwe en these two c ases is that in (b), the two plaquettes fr om which these half-e dges c ome c oincide as in (a), while in (c) they don ’t. Since in total w e ha v e 8 l half-edges, and since at least 4 are needed to form a single lo op, the total num b er of loops is no more than 2 l . Th us, the total factor that they contribute is no more than N 2 l . Putting ev erything together, w e get that the dominant contribution asymptoti- cally to our expression is of order 1 N l . This comes from those terms whic h, when represen ted graphically , con tain only case (a) from Figure 3. Ho w can the latter situation arise? It should be clear that it would only happ en if w e can group the plaquettes into pairs, with one cop y in the pair oriented one wa y , while the other the opp osite wa y , and then couple the half-edges as indicated in Figure 4 b ecause any other scenario w ould pro duce cases (b) or (c). Figure 4: This is a gr aphic al r epr esentation of the only way two plaquettes c an b e c ouple d to gether in or der to solely pr o duc e instanc es of c ase (a) fr om Figur e 3. T o se e this, supp ose the half-e dges ar e c ouple d to pr o duc e c ase (a) for e.g., the b ottom left c orner. It fol lows that the other two half-e dges c oming fr om the b ottom e dge must b e c ouple d to gether as wel l. F ol lowing them ar ound the b ottom right c orner, we c onclude that the c orr esp onding two half-e dges c oming fr om the right e dge must also b e c ouple d to gether, for otherwise, we would have either c ase (b) or (c) at the b ottom right c orner. Pr o c e e ding like this ar ound the plaquette, we c onclude that the two half-e dges c oming fr om these two plaquettes must only c ouple among themselves, giving the pictur e on the right. Since all such terms ev aluate to the same v alue 1 N l , it remains to find the total n umber of such terms. In other words, we m ust determine the total num ber of w ays we can group l plaquettes in to opp ositely oriented pairs. F or an o dd l , this is impossible, and thus we do indeed hav e that the l -th momen t of ρ N ( t ) is O ( 1 N l +1 ) = m l N l + O ( 1 N l +1 ). F or an ev en l , the num b er of such terms is equal to ( l − 1)!! × 2 l 2 × K l 2 ×  D 2  l 2 , where the first factor comes from the n um b er of wa ys one can subdivide a set of l elements in to pairs. The second is a consequence of making a choice of orien tation for eac h plaquette in eac h pair (each plaquette enters with b oth ori- en tations as is clear from (1)). The third is the num b er of sites where the t w o paired plaquettes can b e located. The fourth is the c hoice of the co ordinate plane in space to which the palquettes are parallel. Recalling no w that t has a factor of K in the denominator as well as a factor of 1 2 (coming from the real part), we get that for even l , (4) is equal to 1 (2 K ) l × ( l − 1)!!  2 K  D 2  l 2 × 1 N l + O ( 1 N l +1 ) = 1 N l ( l − 1)!!  2 K ( D 2 )  l 2 + O ( 1 N l +1 ) , whic h is exactly what we w ant, th us completing the pro of. The theorem ab o ve states essentially that dρ N [ t ] ∼ s 2 K N 2 π D ( D − 1) e − 2 K N 2 D ( D − 1) t 2 dt as N → ∞ . (5) Let us th us replace dρ N [ t ] in (2) with the asymptotic Gaussian. Then, w e get the following expression s 2 K N 2 π D ( D − 1) ˆ e 2 N 2 K λ t e − 2 K N 2 D ( D − 1) t 2 dt ∼ e K N 2 D ( D − 1) 2 λ 2 . (6) Ho w reliable is this result as an asymptotic of the partition function? In other w ords, how reliable is the replacement (5) in (2)? Comparing (6) to the known exact asymptotic for D = 2 obtained in [5], we see that the answ er ab o ve coin- cides with the correct expression for λ sufficien tly large ( λ ≥ 2). But, it do es not match with the correct b ehaviour for 0 < λ ≤ 2; notably it do es not detect the present of the phase transition. The reason for this is not hard to see. While the theorem ab o ve justifies the re- placemen t (5) in an in tegral of the form ´ f ( t ) dρ N [ t ], in our case, the in tegrated function not only dep ends on N , it dep ends on it exp onentially and with the same weigh t in N . Put differently , there are tw o pro cesses which comp ete to decide the asymptotic of (2). One of them is the concen tration of measure phe- nomenon whic h is manifested b y the Gaussian, the second term in the in tegrand in (6), trying to set t = 0. The other process is the minimization of the action con tained in the first term in the integrand in (6), which is aiming to make the action as small as p ossible. This corresponds to pushing t to its maximum v alue t =  D 2  . 1 Whic h of these pro cesses wins is decided by their relativ e w eights in the exponents, i.e. by λ . It is th us clear that it is precisely when λ gro ws, that the action term b ecomes less relev ant than the measure one and vice versa. W e can thus expect that the asymptotic replacemen t (5) is reliable in the limit λ → ∞ , whic h is of course the strong-coupling limit, and is unreliable in the limit λ → 0. Alas it is exactly the latter limit which is the physically relev ant one. Nonetheless, as is often the case with asymptotic analysis, the approxima- tions often work m uc h b etter than exp ected. This is exemplified ab ov e in the D = 2 case, where λ ≥ 2 is already “large” and the asymptotic agrees precisely with the exact answer. Let us finish with three remarks. First, note that in view of the discussion abov e, the asymptotic replacemen t (5) can b e though t of as an alternativ e deriv ation of 1 It is not hard to see from (1) that the maximum v alue of t , attained when e.g. all the U ’s are equal to identit y , is  D 2  . the low est order contribution to the strong coupling expansion. Now, how can w e generate the higher orders? T o do so, we should note that strictly sp eaking, (5) do es not follow from the theorem. Rather, one can only conclude that dρ N [ t ] ∼ α N exp  − N 2  c 2 ( N ) t 2 + c 3 ( N ) t 3 + c 4 ( N ) t 4 + . . .   dt where c 2 , c 3 , c 4 , . . . can each b e expanded in p o wers of 1 N , and α N is chosen to normalize the expression on the righ t hand side to 1. T o determine these higher order corrections, one would need to consider the next orders of corrections to the moments calculated ab o ve, and equate them with the asymptotic expansion of the momen ts of the expression on the right ab o ve. This can b e done sys- tematically , as at an y given order, only finitely many new co efficien ts would b e relev ant; How ev er, the graphical calculations rapidly get quite unwieldy . Since there is already a w ell-developed and understo od strong coupling expansion, there is little incentiv e to develop these calculations in this setting. The second remark we would lik e to make concerns the other limit λ → 0. Can one use (2) to calculate the asymptotic to Z in that limit? Alas, as will b e clear b elo w, it do es not lo ok possible to go beyond the lo west order appro ximation. Ho wev er, the argument and its conclusion are quite simple and instructiv e, so they are worth presen ting. Note that if λ → 0, we expect the “action” term to b e the dominan t one. Th us, w e exp ect the main contribution to come from the neigh b ourho od of the p oint that minimizes the action. After taking the pushforward, this corresponds to the maximum p ossible v alue of t which, as men tioned ab o ve, is  D 2  . Ho w do es dρ N [ t ] b ehav e in the neigh b ourho od of this p oint? The exact answ er w ould require finding the Hausdorff measure induced by the Haar measure on the lev el sets of the function defined in (1). This is a difficult undertaking whic h amoun ts to essentially obtaining exactly the partition function as a function N , λ and K . Ho wev er, if one writes the group elements U as e X , where X is in the Lie algebra su ( N ), and keeps only the low est order terms in the expansion of the exp onen tial, the problem simplifies considerably . This is b ecause the equation t ( U ) = constan t becomes a homogeneous one of order 2 if one only k eeps the terms up to the quadratic order. 2 No w, if w e coun t the degrees of freedom, taking in to account gauge in v ariance, w e see that dρ N [ t ] should scale as t ( D − 1)( N 2 − 1) K 2 in the neighbourho o d of t =  D 2  . Therefore, let us replace dρ N [ t ] with this expression in (2). W e get ˆ e 2 N 2 K t λ t ( D − 1)( N 2 − 1) K 2 dt ∼ ˆ e N 2 K ( 2 t λ + ( D − 1) 2 ln t ) dt. The exp onen t is increasing as a function of t . Thus, the dominant contribution comes from the upper endp oint of the supp ort of the integral which is  D 2  . 2 Strictly sp eaking, in our setting, the lo west nonv anishing order after the constant term is not the quadratic but the linear one. How ever, the linear contribution does not scale with N while the quadratic one do es, and th us, is the dominan t one. Alternativ ely , one can recall the standard fact that the large N limits of U ( N ) and S U ( N ) theories coincide and revert to the S U ( N ) case if desired. Therefore, the asymptotic free energy is given b y N 2 K  2 ( D 2 ) λ + D − 1 2 ln  D 2   . Comparing this with the expression obtained in [5], w e see that we do reco ver the low est order contribution to the expression there (which is 2 N 2 K λ ). Unfortu- nately , the metho d ab o ve does not seem to b e adapted to go b ey ond the low est order. The final remark w e w ould like to mak e is to note that the reason that our results are only reliable in the large coupling limit is because, figurativ ely sp eaking, the measure and the action work against each other. One may wonder if there are other settings where they w on’t b e in such a blatan t opp osition, and ma y actu- ally work together to make the asymptotic more reliable in the con tinuum limit instead of less. It turns out that this is indeed the case for the principal chiral mo del, as is rep orted elsewhere [6]. Ac knowledgmen ts: The author would like to thank J. Merhej for reading a preliminary v ersion of this paper and for the n umerous comments which ha ve greatly improv ed its readability . References [1] M. Ledoux, “The concen tration of measure phenomenon”, Math. Surveys Monogr., 89, American Mathematical So ciety , Providence, 2001. [2] B. Collins, S. Matsumoto, J. Nov ak, “The W eingarten Calculus”, Notices Amer. Math. So c. 69 (2022), no. 5, 734–745. [3] D. W eingarten, “Asymptotic b eha vior of group in tegrals in the limit of infinite rank”, J. Mathematical Phys. 19 (1978), no. 5, 999-1001. [4] M. Creutz, “Quarks, Gluons and Lattices”, Cambridge Universit y Press, 2023. [5] D. Gross, E. Witten, “P ossible third-order phase transition in the large-N lattice gauge theory”, Phys. Rev. D 21, (1980), 446–453. [6] T. Tlas, “A Concen tration of Measure Phenomenon in the Principal Chiral Mo del”, to app ear. Department of Mathematics, American University of Beirut, Beirut, Lebanon. Email address : tamer.tlas@aub.edu.lb