math.PR 2026-03-26 0

One-arm probabilities for the two-dimensional metric-graph and discrete Gaussian free field

We study the one-arm probability in the level-set percolation of the discrete and metric-graph Gaussian free field (GFF) defined on a box with Dirichlet boundary conditions. For the metric-graph case, we establish asymptotic estimates on two one-arm …

Authors: Yijie Bi, Yifan Gao, Xinyi Li

ONE-ARM PR OBABILITIES F OR THE TW O-DIMENSIONAL METRIC-GRAPH AND DISCRETE GA USSIAN FREE FIELD YIJIE BI 1 , YIF AN GA O 2 , AND XINYI LI 3 Abstract. W e study the one-arm probabilit y in the level-set percolation of the discrete and metric-graph Gaussian free field (GFF) defined on a b o x with Dirichlet b oundary conditions. F or the metric-graph case, we establish asymptotic estimates on tw o one-arm probabilities of in terest. F or the discrete case, w e sho w up-to-constan ts bounds on the p oint-to-bulk probabilit y and demonstrate its difference from the metric-graph case. 1. Introduction The Gaussian free field (GFF) is the canonical massless Gaussian mo del of a random function on a graph or domain and pla ys a central role in probability , statistical ph ysics, and conformal field theory; its lev el sets (excursion sets) therefore pro vide a natural and ric h model for studying geometric phase transitions and connectivity properties. The study of lev el-set p ercolation of Gaussian free field has a long history; see for instance [ 28 , 6 ]. The long-range nature of the mo del mak es this problem particularly in teresting, but also difficult to analyze. In three and higher dimensions, muc h progress has b een made in the last decade. T o name but a few: finiteness [ 33 ] and strict positivity [ 16 ] of the critical lev el, large deviation estimates of the disconnection probability [ 35 ]; (stretched)-exponential deca y of off- critical one-arm probability [ 31 , 26 ], sharpness of phase transition [ 22 ], etc. The “con tin uum” v ariant of this mo del, namely the metric-graph GFF, exhibits nice in tegrability feature (e.g. the explicit tw o-point function of the sign cluster [ 29 ] and the cluster capacity functional [ 17 ]), whic h allo ws to iden tify the critical lev el, and understand the critical and near-critical behavior of this mo del (e.g. critical exp onen ts [ 18 ], arm probabilities [ 21 , 11 , 20 , 8 , 10 ], cluster v olume [ 19 ], c hemical distance [ 13 ], crossing probability [ 14 ], quasi-m ultiplicativit y [ 9 ], incipient infinite clusters [ 7 ], switching iden tit y [ 36 ]). The lev el sets of GFF in t w o dimensions hav e also b een studied in tensiv ely in recent years. The work [ 12 ] considers chemical distances in terms of bulk annulus-crossing of the lev el set, while [ 14 ] considers left-to-right crossing of the sign clusters for b oth discrete and metric-graph GFF in a rectangular b o x. In [ 2 ], the scaling limit of certain natural interfaces of 2D metric- graph GFF is iden tified as SLE 4 lev el lines. In [ 15 ], upp er and lo w er b ounds are established for the critical threshold with resp ect to the b oundary annulus-crossing even t under the discrete setting. There are also series of works aimed at understanding the tw o-sided lev el sets of 2D GFF [ 25 , 1 , 24 ] and the arm probabilities [ 23 , 3 ]. Despite all the existing literature, the level-set p ercolation of 2D GFF is still quite far from b eing well understo od. In this work, we establish v arious results regarding one-arm probabilities for b oth 2D metric- graph and discrete GFF. F or the metric graph, we obtain the asymptotics of the p oin t-to-bulk probabilit y in Proposition 1.1 . W e also establish an upp er b ound on the p oin t-to-b oundary probabilit y at non-negativ e levels that differs from previously-known asymptotics for negativ e lev els, revealing a phase transition at lev el 0. F or the discrete GFF, we sho w that the p oin t-to- bulk probability is of the same order as the corresp onding metric-graph quan tity , but establish a gap in the pre-factor b et w een the t w o cases. W e w ould like to emphasize that the asymptotics w e established in b oth cases are also explicit in the level h . 1 Sc ho ol of Mathematical Sciences, P eking Universit y , 2200010925@stu.pku.edu.cn . 2 Institute for Theoretical Sciences, W estlake Universit y , gaoyifan75@w estlake.edu.cn . 3 Beijing In ternational Center for Mathematical Research, P eking Universit y , xinyili@bicmr.pku.edu.cn . 1 One-arm probabilities for 2D GFF 2 Notation. W e begin with necessary notation and refer readers to Section 2 for precise defini- tions. F or x ∈ Z 2 , we use | x | ∞ to denote its l ∞ norm. F or any N ∈ N , let B N : = { x ∈ Z 2 : | x | ∞ ≤ N } b e the b o x cen tered at 0 with side length 2 N , and let e B N denote its metric-graph v ersion obtained by joining each pair of neighboring vertices with a line segmen t of length 2, fol- lo wing the con v en tion of [ 29 ]. W e use ∂ i B N and ∂ B N to denote the inner and outer b oundaries of B N , resp ectiv ely . Let e φ N b e the GFF on the metric-graph e B N +1 with Diric hlet b oundary conditions (i.e., the v alues of e φ N on ∂ i B N +1 and line segmen ts joining vertices in it are pinned to 0). Let φ N denote the discrete GFF on B N +1 , defined by the restriction of e φ N to B N +1 . F or all h ∈ R , w e denote the level set of φ N b y E ≥ h N : = { x ∈ B N : φ N ,x ≥ h } . F or all non-empty A 1 , A 2 ⊂ Z 2 , let { A 1 φ N ≥ h ← → A 2 } b e the even t that there exists a nearest-neigh bor path in E ≥ h N joining A 1 and A 2 . W e simply write { z φ N ≥ h ← → A 2 } if A 1 is the singleton { z } . Throughout, the following notation will b e hea vily used. F or tw o p ositiv e sequences f N and g N , we write f N ≲ g N if there exists some universal constant c > 0 such that f N ≤ cg N for all N ∈ N . Define f N ≳ g N if f − 1 N ≲ g − 1 N , and write f N ≍ g N if f N ≲ g N and f N ≳ g N . F or any a, b > 0, define a ∇ b : = ( a ∨ b ) ∧ 1 . Results. W e start with a preliminary result for the metric-graph GFF e φ N that provides up-to- constan ts estimates for the one-arm probabilities in the bulk at any lev el h ∈ R . Prop osition 1.1. The fol lowing estimates hold: (1.1) P [0 e φ N ≥ h ← → ∂ B N/ 2 ] ≍ | h | √ log N ∇ 1 √ log N , h ≤ 0; (1.2) e − a 1 h 2 (log N ) − 1 2 ≲ P [0 e φ N ≥ h ← → ∂ B N/ 2 ] ≲ e − a 2 h 2 (log N ) − 1 2 , h > 0 . A b ove, a 1 , a 2 > 0 ar e some universal c onstants. W e refer to Prop osition 3.1 for a stronger result, which considers connection to ∂ B rN for an y r ∈ (0 , 1 2 ) and provides similar estimates given the v alue of e φ N at the origin. Note that in ( 1.1 ) and ( 1.2 ), the decay in N is alwa ys (log N ) − 1 2 for all h ∈ R . Ho w ever, as w e no w sho w, the deca y rate for the one-arm probabilit y to the b oundary satisfies a phase transition across lev el 0. In fact, when h < 0, the “boundary” one-arm probability can be exactly determined by using [ 30 ]. More precisely , by [ 30 , Prop osition 2], (1.3) P [0 e φ N ≥ h ← → ∂ B N ] = P [ | e φ N , 0 | ≤ | h | ] for all h < 0 . Note that e φ N , 0 is a cen tered Gaussian random v ariable with v ariance given by the Green’s function e G N (0 , 0) ≍ log N (see Sec tion 2.2 ). Hence, from ( 1.3 ), one has (1.4) P [0 e φ N ≥ h ← → ∂ i B N ] ≍ P [0 e φ N ≥ h ← → ∂ B N ] ≍ | h | (log N ) − 1 2 for all h < 0 . In order to get a non-trivial arm probability when h > 0, w e will consider the connection to ∂ i B N instead of ∂ B N . In comparison, when h > 0, the boundary one-arm probabilit y deca ys at least p olynomially fast in N . Prop osition 1.2. Ther e exists a c onstant c such that for al l h ≥ 0 and al l N , (1.5) P [0 e φ N ≥ h ← → ∂ i B N ] ≲ (log N ) − 1 N − ch 2 . While this pap er was b eing written, w e w ere informed that Pierre-F ran¸ cois Ro driguez and W en Zhang were considering similar problems in the massive metric-graph setting, where the b oundary conditions are replaced by a killing with rate of order N − 2 on the metric-graph Bro wnian motion. In their recent preprin t [ 34 ], they implemen ted a renormalization argumen t to prov e an estimate on the one-arm probability for sub critical h analogous to ( 1.2 ), and obtained the precise v alue of the pre-factor b efore h 2 (analogous to a 1 , a 2 in ( 1.2 )) which tak es the form of some version of the Brownian capacit y of a line segment in the plane. Y. Bi, Y. Gao, X. Li 3 W e now turn to the main res ult of this work, namely the “bulk” one-arm probability for the discrete GFF φ N . W e sho w that its asymptotics in N remains of order (log N ) − 1 2 as in the metric graph case. How ev er, the discrete estimates requires muc h more efforts to establish due to issues from the discrete nature of the problem, suc h as that the exploration process asso ciated with the level set will not stop exactly at the level h . This will b e more clear in Section 4 . Theorem 1.3. We have the fol lowing estimates: (1.6) P [0 φ N ≥ h ← → ∂ B N/ 2 ] ≍ | h | √ log N ∇ 1 √ log N , h ≤ 0; (1.7) e − a 3 h 2 (log N ) − 1 2 ≲ P [0 φ N ≥ h ← → ∂ B N/ 2 ] ≲ e − a 4 h 2 (log N ) − 1 2 , h > 0 , wher e a 3 , a 4 > 0 ar e some universal c onstants. Moreo v er, we pro ve that the ab o ve one-arm probabilit y differs from its coun terpart in the metric graph by a constan t m ultiple of (log N ) − 1 2 . This result b ears some resemblance to [ 14 , Theorem 1.2], where it is prov ed that the left-right crossing probabilities of the sign cluster of the discrete and metric-graph GFF’s differ b y a p ositiv e constant. Theorem 1.4. F or any h ∈ R , ther e exists some c = c ( h ) > 0 such that (1.8) P [0 φ N ≥ h ← → ∂ B N/ 2 ] − P [0 e φ N ≥ h ← → ∂ B N/ 2 ] ≥ c (log N ) − 1 2 . As an intermediate step in the pro of of ( 1.8 ), we also establish the follo wing RSW-type b ounds, whic h is of indep enden t interest: with p ositiv e probability uniformly in N , there exists a circuit (i.e., lo op) ab o v e any fixed level in an y annulus A αN ,β N : = B β N \ B αN for 0 < α < β < 1. Prop osition 1.5. F or al l h ∈ R and 0 < α < β < 1 , ther e exists c = c ( h, α, β ) > 0 such that for al l lar ge N , (1.9) P [ ther e exists a cir cuit in A αN ,β N that surr ounds B αN , on which φ N ≥ h ] > c, and henc e (1.10) P [ B αN φ N ≤ h ← → ∂ B β N ] < 1 − c. Note that ( 1.10 ) confirms the sp eculation in [ 12 , Remark 1]. Inciden tally , our results indicate the following b ound on the c hemical distance on the discrete lev el set across a macroscopic annulus given connectivit y , parallel to the result on the metric graph [ 13 , Theorem 1]). F or h ∈ R , let D N ,h denote the graph distance on the level set E ≥ h N . F or any h ∈ R , 0 < α < β < γ < 1, and ε > 0, there exists a constan t c such that (1.11) lim sup N →∞ P [ D N ,h ( B αN , ∂ B β N ) > cN (log N ) 1 / 4 | B αN φ N ≥ h ← → ∂ B γ N ] ≤ ε. Note that such a result has already b een sp eculated in [ 13 , Section 1.3]. Pro of strategy . W e no w briefly outline the main ideas of the pro ofs. The cen tral to ol through- out the pap er is the exploration martingale, in troduced in [ 13 ]. In b oth the metric-graph and discrete settings, we construct a contin uous-time exploration pro cess of the level-set cluster starting from the origin. Along this exploration, we track the evolution of the GFF on a finite set of vertices and consider the asso ciated martingale given b y the conditional exp ectation of a suitable observ able, conditioned on the current explored b oundary . This approach allows us to reform ulate the one-arm ev en t in terms of a constraint on the tra jectory of the exploration martingale: namely , the ev en t that the martingale remains ab ov e a certain threshold o ver some time in terv al. This probabilistic reformulation is more amenable to analysis thanks to the Dubins-Sc hw arz Theorem (see Lemma 2.4 ), and enables us to derive sharp estimates. The metric-graph case is comparativ ely simpler. In this setting, the exploration process enjo ys an exact stopping property: the exploration terminates precisely when all b oundary p oin ts of One-arm probabilities for 2D GFF 4 the cluster simultaneously reac h the lev el h (see ( 3.4 )). This structural feature allo ws for a relativ ely direct analysis of the asso ciated martingale. In con trast, the discrete setting lac ks such an exact stopping mechanism. T o ov ercome this difficult y , we adapt an entropic repulsion argument from [ 14 ] (see Prop osition 4.2 ). Roughly sp eaking, we sho w that with high probability , the harmonic av erage of the GFF along the b oundary of the explored cluster do es not fall significan tly b elo w the level h . This substitute con trol compensates for the absence of the exact stopping property and allo ws us to deriv e an upp er b ound on the discrete one-arm probabilit y via a similar martingale argument. The corresp onding low er bound follo ws more directly , as the discrete one-arm even t is implied b y its metric-graph counterpart. Organization. The rest of the pap er is organized as follo ws. In Section 2, w e in tro duce the no- tation used in this paper, together with some key to ols (in particular the exploration martingale in tro duced in [ 13 ], from which we hav e drawn a lot of inspiration, and the connection b etw een (metric-graph) GFF level sets, the lo op soup and the excursion cloud, established in [ 29 ] and [ 2 ]). W e will also establish some preliminary results in Section 2. In particular, we pro ve a stronger version of Prop osition 1.5 (c.f. Prop osition 2.7 ). In Section 3, we prov e the results on the metric-graph GFF, namely Prop ositions 1.1 and 1.2 . Finally , in Section 4, w e fo cus on the discrete case, establishing Theorems 1.3 and 1.4 and sketc hing a pro of of ( 1.11 ). 2. Preliminaries F or v = ( v 1 , v 2 ) ∈ R 2 , let | v | ∞ : = | v 1 | ∨ | v 2 | and | v | : = p v 2 1 + v 2 2 b e its l ∞ and l 2 norms respec- tiv ely . F or any nonempt y A 1 , A 2 ⊂ R 2 , define the l ∞ -distance b et w een them by dist( A 1 , A 2 ) : = inf {| x 1 − x 2 | ∞ : x 1 ∈ A 1 , x 2 ∈ A 2 } , and write dist( x, A 2 ) if A 1 = { x } is a singleton. F or all x, y ∈ Z 2 and r > 0, w e say that x, y are nearest neighbors and write x ∼ y if | x − y | = 1. W e define the discrete r -neighborho o d of x b y B ( x, r ) : = { y ∈ Z 2 : | x − y | ∞ ≤ r } , and write B r = B (0 , r ). F or an y 0 < r < r ′ < ∞ , define A r,r ′ = B r ′ \ B r . F or A ⊂ Z 2 , de- fine its inner b oundary by ∂ i A = { x ∈ A : ∃ y ∈ A c , x ∼ y } , and its outer b oundary by ∂ A = { x ∈ A c : ∃ y ∈ A, x ∼ y } . F or an y non-empty A ⊂ Z 2 , let e A denote its metric graph ver- sion which is obtained b y joining eac h pair of neighboring vertices by a line segmen t of length 2. The unlab eled constants c, c ′ , c ′′ , etc. may v ary from line to line, whereas the lab eled constants c i , c ′ i etc. are fixed throughout eac h se ction . Unless stated otherwise, all constants are univ ersal and do not rely on any additional parameters. 2.1. Basic results on Brownian motions and martingales. W e start with an useful esti- mate for standard one-dimensional Brownian motions. Let (2.1) Φ( s ) : = Z ∞ s 1 √ 2 π e − 1 2 u 2 du denote the tail probability of the standard normal distribution. W e will use the following es- timate on the hitting probability of a line with an y given slop e; see e.g. [ 4 , equation (2.0.2) in P art I I] or [ 13 , Prop osition 15]. Lemma 2.1. Supp ose ( B t ) t ≥ 0 is a standar d one-dimensional Br ownian Motion. F or m ∈ R and b > 0 , let τ = inf { t > 0 : B t ≤ mt − b } . F or any T > 0 , P [ τ ≤ T ] = 1 − ψ ( m, b, T ) , wher e (2.2) ψ ( m, b, T ) : = Φ  m √ T − b √ T  − e 2 bm Φ  m √ T + b √ T  . F or some constan ts c, c ′ > 0, the following asymptotics hold: (2.3) be − cm 2 T √ T ≲ ψ ( m, b, T ) ≲ be − c ′ m 2 T √ T , when m > 0 , 0 < b ≤ √ T ; Y. Bi, Y. Gao, X. Li 5 (2.4) ψ ( m, b, T ) ≍ | mb |∇ b √ T , when m ≤ 0 . Next, w e turn to the Brownian motion on the metric graph. The Brownian motion X can b e defined on e Z 2 in a certain w a y suc h that it b eha v es like a standard Brownian motion in the in terior of edges and “chooses to mak e excursions on each incoming edge uniformly at random” on v ertices (see [ 29 , Section 2] for a rigorous definition). F or x ∈ e Z 2 , w e denote the la w of X with X 0 = x b y P x and the corresp onding exp ectation by E x . F or all non-empt y A ⊂ e Z 2 , the first hitting time of A by X is given b y τ A : = inf { t ≥ 0 : X t ∈ A } . In [ 29 , Section 2], Lupu also introduced the Green’s function for the Bro wnian motion on e Z 2 killed up on exiting some subset A of e Z 2 . W e denote it by e G A ( x, y ), for all x, y ∈ A , and abbreviate e G N : = e G e B N +1 . F or V ⊂ Z 2 non-empt y , let G V denote the Green’s function for the simple random walk on Z 2 killed up on exiting V . Then, it is argued in [ 29 ] that G V is the restriction of e G ^ V ∪ ∂ V to V . F or A ⊂ e B N +1 , and x ∈ e B N +1 , y ∈ A , define the harmonic measure of A from x relative to e B N +1 at y b y (2.5) H N ( x, y ; A ) = P x [ X τ A ∪ ( e B N +1 ) c = y ] , and write H N ( U, y ; A ) = P x ∈ U H N ( x, y ; A ) for all U ⊂ B N . F or an y subset D ⊂ A , write H N ( U, D ; A ) = X y ∈ D H ( U, y ; A ) , where the equalit y is w ell defined b ecause the n um b er of y ’s suc h that H ( U, y ; A ) > 0 is obviously finite. F or simplicit y , we write H N ( U, A ) = H N ( U, A ; A ), and write H N ( x, A ) = H N ( { x } , A ) for the harmonic measure of A relative to e B N +1 seen from x . Finally , w e set the av erage harmonic measure from U as follows: (2.6) H N ( U, · ) = 1 | U | H N ( U, · ) . W e will frequen tly use the following v ersion of Beurling estimate stated for Brownian motion on the metric graph. Lemma 2.2. Ther e exists c > 0 such that for any c onne cte d A ⊂ Z 2 with 0 ∈ A and A ∩ ∂ B n  = ∅ , we have P x [ τ ∂ B n < τ A ] ≤ cn − 1 2 . In fact, Lemma 2.2 follows directly from the corresp onding result for simple random walk (e.g. [ 27 , Theorem 6.8.3]), since X restricted to Z 2 can b e viewed as a simple random w alk. Moreo v er, by [ 27 , Prop osition 6.4.1], for all 0 < k < N and x ∈ A k,N , we ha v e H N ( x, B k ) = log N − log | x | ∞ + O ( k − 1 ) log N − log k . Com bining this estimate and Lemma 2.2 , one immediately obtains Lemma 2.3. Ther e exist c, c ′ > 0 such that for any x ∈ e A 5 8 N , 7 8 N and S with 0 ∈ S ⊂ e B N/ 2 , we have c (log N − log diam( S )) − 1 ≤ H N ( x, S ) ≤ c ′ (log N − log diam( S )) − 1 , wher e diam( S ) : = sup x,y ∈ S | x − y | ∞ . Finally , w e cite a version of the Dubins-Sc h warz Theorem [ 32 , Theorem 1.7 in Chapter 5], whic h will b e useful in estimating deviation probabilities of exploration martingales (see Sec- tion 2.3 ). F or any con tin uous martingale M , let ⟨ M ⟩ . denote its quadratic v ariation pro cess. One-arm probabilities for 2D GFF 6 Lemma 2.4. L et M b e a c ontinuous martingale, T t = inf { s : ⟨ M ⟩ s > t } , and W b e the fol lowing pr o c ess: W t = ( M T t − M 0 , t < ⟨ M ⟩ ∞ ; M ∞ − M 0 , t ≥ ⟨ M ⟩ ∞ . Then W is a standar d Br ownian motion stopp e d at ⟨ M ⟩ ∞ . 2.2. The Gaussian free field (GFF). W e refer to [ 29 ] for a more comprehensive in tro duction. Define e φ N as the metric-graph GFF on e B N +1 with Dirichlet b oundary conditions, i.e., the cen tered contin uous Gaussian field on e B N +1 with co v ariance function e G N and b oundary v alues 0 on ∂ i B N +1 . Define the corresp onding discrete GFF φ N b y the restriction of e φ N to B N +1 . F or h ∈ R , define the level set of e φ N ab o v e level h b y e E ≥ h N = { x ∈ e B N +1 : e φ N ,x ≥ h } , and set E ≥ h N = e E ≥ h N ∩ Z 2 . W e define the lev el sets b elow h b y e E 0 } , and define F I = σ { φ N ,x , x ∈ I } . F or any random v ariable X , sigma algebra F and even t A in the same probability space, and an y k ∈ N , define the conditional k -th (cen tral) momen t of X giv en A and F b y (2.8) µ ( k ) [ X | A, F ] : = E h   X − E [ X | A, F ]   k    A, F i . The following lemma con trols moments of the harmonic av erage of the GFF on D ( I ), which follo ws from the results in [ 14 ]. Lemma 2.5. F or I ⊂ B N/ 2 , let I + and I − b e a p artition of I satisfying that for al l x ∈ I − , ther e is some x ′ ∈ I + such that x ∼ x ′ . F or U ⊂ A 5 8 N , 7 8 N , define Y : = X x ∈ D ( I ) H ( U, x ; I ) φ N ,x . F or any h ∈ R , define the event A = { φ N ,x < h : x ∈ I − } ∩ { φ N ,x ≥ h : x ∈ I + } . Then, for any p ositive inte ger k , (2.9) µ (2 k ) [ Y | A, F I + ] ≤ (2 k − 1)!! · 16 k ξ ( U, I ) k H N ( U, I ) 2 k , wher e (2.10) ξ ( U, I ) : = sup x ∈ I \ D ( I ) H N ( U, x ; I \ D ( I )) /H N ( U, I ) . Pr o of. By [ 14 , Corollary 4.4], we obtain V ar[ Y | F I + ] ≤ 16 ξ ( U, I ) · H N ( U, I ) 2 . Since the conditional la w of Y giv en F I + is still Gaussian (Mark ov property of GFF), we ha v e µ (2 k ) [ Y | F I + ] ≤ (2 k − 1)!! · 16 k ξ ( U, I ) k H N ( U, I ) 2 k . Y. Bi, Y. Gao, X. Li 7 The lemma then follo ws from the fact that µ (2 k ) [ Y | A, F I + ] ≤ µ (2 k ) [ Y | F I + ] , whic h can be deduced from the Brascamp-Lieb inequalit y [ 5 , Theorem 5.1], similar to the pro of of [ 14 , Lemma 4.3] (we refer the reader to it for details). □ 2.3. Exploration martingales. W e no w briefly recall the exploration martingales associated with lev el sets of GFF, whic h serves a key to ol in [ 12 , 13 , 14 ]. F or S ⊂ B N , w e define the exploration of the metric-graph level set e E ≥ h N , ( e I t ) t ≥ 0 , with source e I 0 = S b y e I t = { u ∈ e B N +1 : e D N ,h ( u, S ) ≤ t } . In the discrete case, we will use ( I t ) t ≥ 0 to denote the exploration pro cess associated with the discrete lev el set E ≥ h N and source S , whic h is defined as follo ws. W e b egin by setting V 0 = S , A 0 = V 0 ∩ E ≥ h N and B 0 = V 0 ∩ E 0 is defined through linear interpolation. Hence, I t is indeed a metric graph for an y t . F or U ⊂ e B N finite, we define the observ ables X U and X U b y (2.11) X U : = X x ∈ U e φ N ,x and X U : = X U | U | . Then, we define the exploration martingales corresp onding to observ ables X U b y (2.12) M U,t : = E [ X U | F e I t ] in the metric-graph case, and M U,t : = E [ X U | F I t ] in the discrete case. Here, with slight abuse of notation we use the same sym b ol for both cases, but it would be clear from the context. W e similarly define the renormalized exploration martingale M U b y replacing X U with X U in the definitions ab o v e. 2.4. Isomorphism theorem and circuit estimates. In this section, w e first recall a pow erful isomorphism theorem for GFF from [ 2 ], and then use it to pro v e Prop osition 1.5 . W e refer readers to [ 2 , Section 2] for rigorous introduction of Brownian lo op soups and Bro wnian excursions on metric graphs. Let e L 1 / 2 N b e the lo op soup in e B N +1 with intensit y 1 2 . F or h > 0, let Ξ h N b e the excursion cloud in e B N +1 with in tensity h 2 2 , whic h is a random set of Bro wnian excursions in e B N +1 with endp oints in ∂ e B N +1 . W e consider all the clusters made by lo ops and excursions inside indep enden t union of e L 1 / 2 N and Ξ h N . Let e A − h N b e the union of ∂ e B N +1 and topological closures of the clusters that contain at least one excursion (i.e. are connected to ∂ e B N +1 ). According to [ 2 , Prop ositions 2.4 and 2.5], w e hav e the following description of level sets of metric-graph GFF. Lemma 2.6. L et h > 0 . Ther e is a c oupling such that the clusters of lo ops and excursions (in the union of e L 1 / 2 N and Ξ h N ) c orr esp ond to the sign clusters of e φ N + h in the fol lowing way. • The zer o set of e φ N + h is exactly the set of vertic es not visite d by any lo op or excursion. • e A − h N is the set of vertic es that c an b e c onne cte d to ∂ e B N +1 via a c ontinuous p ath in e E ≥− h N . • F or those clusters that c ontain no excursion, e ach has e qual pr ob ability to b e a cluster in e E > − h N or e E < − h N , which do es not interse ct ∂ e B N +1 . With Lemma 2.6 , we are no w ready to sho w a v ersion of Proposition 1.5 on the metric graph. As w e will see immediately , it implies the discrete one directly . F or 1 ≤ k < l ≤ N and h ∈ R , let e C h N ,k,l denote the ev en t that there exists a contin uous circuit inside the annulus e A k,l that surrounds e B k , on which e φ N ≥ h . One-arm probabilities for 2D GFF 8 Prop osition 2.7. F or al l h ∈ R and ε ∈ (0 , 1) , ther e exists c = c ( ε, h ) > 0 such that for al l (1 + ε ) k < l ≤ (1 − ε ) N with k sufficiently lar ge, we have P [ e C h N ,k,l ] ≥ c . Pr o of. W e rely on the lo op-soup description of lev el sets of metric-graph GFF, as explained in Lemma 2.6 . Note that we only need to prov e the result for h > 0 by monotonicit y , which we assume tacitly below. Recall the notation in tro duced ab o v e Lemma 2.6 . Let H N ,k,l b e the even t that there exists a loop of e L 1 / 2 N inside e A k,l that surrounds e B k . By [ 3 , Lemma 4.6], there exists c 1 = c 1 ( ϵ ) > 0, (2.13) P [ H N ,k,l ] ≥ c 1 . Note that with the help of the coupling in Lemma 2.6 , it follows that P [ e C h N ,k,l ] ≥ 1 2 P [ e A − h N ∩ e B (1 − ϵ ) N = ∅ , H N ,k,l ] , (2.14) for the following reason: On the ev en t { e A − h N ∩ e B (1 − ϵ ) N = ∅} ∩ H N ,k,l , let γ b e any Bro wnian lo op in e A k,l that surrounds e B k , then the cluster of γ contains no excursion and has probabilit y 1 2 to b e a cluster in e E ≥ h N under the coupling (this explains the prefactor 1 2 app earing in ( 2.14 )), and therefore the lo op γ pro vides a contin uous circuit inside e A k,l ∩ e E ≥ h N that surrounds e B k . Altogether, this implies ( 2.14 ) by using the symmetry of e φ N . Moreo v er, the ev ent { e A − h N ∩ e B (1 − ε ) N = ∅} is indep endent of lo ops inside e B l , and hence of H N ,k,l . Hence, by ( 2.13 ) and ( 2.14 ) it follows that (2.15) P [ e C h N ,k,l ] ≥ c 1 2 P [ e A − h N ∩ e B (1 − ϵ ) N = ∅ ] . It remains to give a low er b ound for P [ e A − h N ∩ e B (1 − ϵ ) N = ∅ ]. F or this, by [ 2 , Lemma 4.13], there exist c 2 = c 2 ( ϵ ) > 0 and δ = δ ( ϵ ) < ε suc h that with probabilit y at least c 2 , there do es not exist a cluster in e L 1 / 2 N that in tersects b oth e B (1 − ϵ ) N and ∂ e B (1 − δ ) N . Moreo v er, the total mass of excursions in e B N +1 that intersect ∂ e B (1 − δ ) N is b ounded from ab o v e uniformly in N (but dep ending on ϵ and h ). Hence, with probabilit y at least c 3 ( ϵ, h ), no excursion reac hes ∂ e B (1 − δ ) N . With ev erything combined, we conclude that P [ e A − h N ∩ e B (1 − ϵ ) N = ∅ ] ≥ c 2 c 3 and hence P [ e C h N ,k,l ] ≥ c 1 c 2 c 3 2 , as desired. □ Pr o of of Pr op osition 1.5 . Noting that a contin uous circuit in e E ≥ h N alw a ys con tains a discrete circuit in E ≥ h N , Prop osition 1.5 follows b y taking k = αN , l = β N and ε < (1 − β ) ∧ β − α α in Prop osition 2.7 . □ 3. The Metric-Graph Case This section is dedicated to the pro of of Propositions 1.1 and 1.2 concerning bulk and bound- ary one-arm probabilities on the metric graph, respectively . W e start with the pro of of Prop o- sition 1.1 , whic h will b e a direct consequence of the follo wing stronger estimate for the one-arm probabilit y giv en the v alue of GFF at the origin. T o state the result, for an y h ∈ R , x > 0, and r ∈ (0 , 1 / 2], we define (3.1) g h N ,r ( x ) : = P h 0 e φ N ≥ h ← → ∂ B rN | e φ N , 0 = h + x p log N i , where we recall that e φ N is the GFF on the metric graph e B N +1 with Dirichlet boundary condi- tions. Prop osition 3.1. F or any h ≤ 0 , x > 0 , and r ∈ (0 , 1 / 2] , we have (3.2) g h N ,r ( x ) ≍ x | h | √ log N ∇ x | log r | 1 2 √ log N . Y. Bi, Y. Gao, X. Li 9 Mor e over, ther e exist c, c ′ , c ′′ > 0 such that if h > 0 , x ≤ c ′′ √ log N | log r | − 1 2 , setting K = K ( h, r ) : = h | log r | − 1 2 , we have (3.3) xe − cK 2 s | log r | log N ≲ g h N ,r ( x ) ≲ xe − c ′ K 2 s | log r | log N . Pr o of. W e follo w a similar strategy as in the pro of of [ 13 , Lemma 24] b y analyzing the corre- sp onding exploration martingale. How ev er, we will k eep trac k of the dep endence on h, x, r more carefully . Precisely , w e consider the exploration pro cess ( e I t ) t ≥ 0 with source e I 0 = { 0 } on e E ≥ h N . Let U = ∂ B ⌊ 3 4 N ⌋ , and consider the exploration martingale M = M U in tro duced in ( 2.12 ). Let σ = inf { t ≥ 0 : e I t ∩ B rN  = ∅} , then { 0 e φ N ≥ h ← → ∂ B rN } = { σ < ∞} . Recall the definition of H N from ( 2.6 ) and observ e that almost surely , (3.4) M t > hH N ( U, e I t ) for all 0 ≤ t < σ ; M ∞ = hH N ( U, e I ∞ ) when σ = ∞ . By Lemma 2.3 , there exist univ ersal constants c 1 , c ′ 1 > 0 suc h that on the ev ent { σ < ∞} , w e ha v e (3.5) c 1 | log r | − 1 ≤ H N ( U, e I σ ) ≤ c ′ 1 | log r | − 1 . Define σ 1 : = inf { t ≥ 0 : H N ( U, e I t ) ≥ c 1 | log r |} , and similarly define σ ′ 1 with c 1 replaced by c ′ 1 . By ( 3.5 ), σ 1 ≤ σ ≤ σ ′ 1 . Therefore, (3.6) { σ < ∞} = { σ < ∞} ∩ { M t > hH N ( U, e I t ) , ∀ 0 ≤ t < σ } ⊂{ σ 1 < ∞} ∩ { M t > hH N ( U, e I t ) , ∀ 0 ≤ t < σ 1 } = : E . Similarly , (3.7) { σ < ∞} ⊃ E ′ : = { σ ′ 1 < ∞} ∩ { M t > hH N ( U, e I t ) , ∀ 0 ≤ t < σ ′ 1 } . W rite P x : = P [ · | e φ N , 0 = h + x √ log N ]. W e now estimate the probabilities of E and E ′ under P x , whic h, combined with the ab o v e inclusions, yield desired b ounds on g h N ,r ( x ) = P x [ σ < ∞ ]. Note that for 0 ≤ s < t ≤ σ , ⟨ M ⟩ t − ⟨ M ⟩ s = V ar( X U |F e I s ) − V ar( X U |F e I t ) = 1 | U | 2 X x,y ∈ U  e G ∂ B N +1 ∪ e I s ( x, y ) − e G ∂ B N +1 ∪ e I t ( x, y )  = 1 | U | X y ∈ U X z ∈ e I t H N ( U, z ; e I t ) e G ∂ B N +1 ∪ e I s ( z , y ) ≍ X z ∈ e I t H N ( U, z ; e I t ) H N ( z , ∂ B N +1 ; e I s ) = H N ( U, e I t ) − H N ( U, e I s ) , where in the “ ≍ ” of the fourth line we used the following estimate: 1 | U | X y ∈ U e G ∂ B N +1 ∪ e I s ( z , y ) ≍ 1 N X y ∈ U H N ( z , y ; U ∪ e I s ) X w ∈ U e G ∂ B N +1 ∪ e I s ( y , w ) ≍ H N ( z , U ; U ∪ e I s ) ≍ H N ( z , ∂ B N +1 ; e I s ) . (See [ 13 , Lemma 20] for a similar calculation.) Hence, there exist c 2 , c ′ 2 > 0 such that for all 0 ≤ s < t ≤ σ , (3.8) c 2 ( ⟨ M ⟩ t − ⟨ M ⟩ s ) ≤ H N ( U, e I t ) − H N ( U, e I s ) ≤ c ′ 2 ( ⟨ M ⟩ t − ⟨ M ⟩ s ) . Moreo v er, by Lemma 2.3 , there exist c 3 , c ′ 3 > 0 such that (3.9) c ′ 3 log N ≤ H N ( U, e I 0 ) ≤ c 3 log N . One-arm probabilities for 2D GFF 10 W e now pro ve the main claims. W e first assume h > 0 and prov e ( 3.3 ). F or this pur- p ose, we deal with the upp er b ound of P x [ E ] first. Note that under P x , we can write M 0 = H N ( U, e I 0 ) e φ N , 0 = H N ( U, e I 0 ) ( h + x √ log N ) by the Marko v prop ert y of GFF. Com bining ( 3.8 ) and ( 3.9 ) with the definition of the even t E from ( 3.6 ), we get (3.10) M t − M 0 > h ( H N ( U, e I t ) − H N ( U, e I 0 )) − x p log N · H N ( U, e I 0 ) ≥ c 2 h ⟨ M ⟩ t − c 3 x √ log N for all 0 ≤ t < σ 1 . Let ( B t ) t ≥ 0 b e a standard Brownian motion. Applying Lemma 2.4 , we ha v e P x [ E ] ≤ P x  M t − M 0 > c 2 h ⟨ M ⟩ t − c 3 x √ log N , 0 ≤ t < σ 1  = P  B t > c 2 ht − c 3 x √ log N , 0 ≤ t < c 1 | log r | − 1  = ψ ( m, b, T ) , where ψ ( · ) was in tro duced in ( 2.2 ), m = c 2 h , b = c 3 x √ log N and T = c 1 | log r | − 1 . Moreo v er, there exists c > 0 small enough suc h that if x ≤ c √ log N | log r | − 1 2 , then b = c 3 x √ log N ≤ cc 3 | log r | − 1 2 ≤ cc 3 c − 1 2 1 √ T ≤ √ T , and by ( 2.3 ), we obtain that ψ ( m, b, T ) ≲ be − c ′ m 2 T √ T ≍ xe − c ′′ K 2 s | log r | log N for some c ′ > 0 and c ′′ = c ′ c 1 c 2 2 . The upp er b ound of ( 3.3 ) no w follows since g h N ,r ( x ) = P x [ σ < ∞ ] ≤ P x [ E ] by ( 3.6 ). The deriv ation for the low er b ound of P x [ E ′ ] is similar. Now, in order for the even t E ′ to o ccur, w e only need to ha v e σ ′ 1 < ∞ and M t − M 0 > h ( H N ( U, e I t ) − H N ( U, e I 0 )) − x √ log N · H N ( U, e I 0 ) holds for all 0 ≤ t < σ ′ 1 , whic h further con tains the even t M t − M 0 > c ′ 2 h ⟨ M ⟩ t − c ′ 3 x √ log N b y ( 3.8 ) and ( 3.9 ). Th us, for some c > 0 (not necessarily the same as the previous one) and x ≤ c √ log N | log r | − 1 2 , we ha v e g h N ,r ( x ) ≥ P x [ E ′ ] ≥ P x  M t − M 0 > c ′ 2 h ⟨ M ⟩ t − c ′ 3 x √ log N , 0 ≤ t < σ ′ 1  = P  B t > c ′ 2 ht − c ′ 3 x √ log N , 0 ≤ t < c ′ 1 | log r | − 1  ≳ xe − c ′ K 2 s | log r | log N for some c ′ > 0, where we used Lemma 2.1 and ( 2.3 ) in the last line. This completes the pro of of ( 3.3 ). W e finally turn to ( 3.2 ), which deals with the case h ≤ 0. In fact, this is similar to the ab ov e. F or an illustration, to derive the upp er b ound, we can simply replace ( 3.10 ) b y M t − M 0 > c ′ 2 h ⟨ M ⟩ t − c 3 x √ log N , and conclude with the following b ound g h N ,r ( x ) ≤ P x [ E ] ≤ P  B t > c ′ 2 ht − c 3 x √ log N , 0 ≤ t < c 1 | log r | − 1  . Using Lemma 2.1 again, and taking ( 2.4 ) into consideration, w e can upper b ound the right-hand side as required. Since the low er b ound of ( 3.2 ) is also similar, we omit its pro of and conclude the pro of of Prop osition 3.1 . □ Next, we use Prop osition 3.1 to finish the pro of of Prop osition 1.1 . Pr o of of Pr op osition 1.1 . Noting that when h < − √ log N , we ha v e the trivial b ound P [0 e φ N ≥ h ← → ∂ B N/ 2 ] ≥ P  0 e φ N ≥− √ log N ← − − − − − − − → ∂ B N/ 2  , Y. Bi, Y. Gao, X. Li 11 hence we ma y assume h ≥ − √ log N without loss of generality . Letting (3.11) ξ h N : = ( e φ N , 0 − h ) / p log N , w e can rewrite the one-arm probabilit y in the following w a y (3.12) P [0 e φ N ≥ h ← → ∂ B N/ 2 ] = E [ g h N , 1 2 ( ξ h N ) 1 ξ h N > 0 ] . Since e φ N , 0 is a centered Gaussian with v ariance of order log N , ξ h N is Gaussian with mean − h √ log N ≤ 1 and v ariance of constant order. By Prop osition 3.1 , there are c, c ′ > 0 such that g h N , 1 2 ( x ) ≲ xη h,c ′ N , ∀ x ∈ [0 , c p log N ] , where (3.13) η h,ρ N : =    | h |∨ 1 √ log N , h ≤ 0; (log N ) − 1 2 e − ρh 2 , h > 0 . Note that there exists a > 0 such that P [ ξ h N > c √ log N ] ≤ N − a . Therefore, E [ g h N , 1 2 ( ξ h N ) 1 ξ h N > 0 ] ≤ E [ g h N , 1 2 ( ξ h N ) 1 0 <ξ h N ≤ c √ log N ] + N − a ≲ E [ ξ h N η h,c ′ N 1 0 <ξ h N ≤ c √ log N ] + N − a ≲ η h,c ′ N + N − a , yielding the desired upp er b ound (b y adjusting c ′ if necessary). F or the low er b ound, w e apply Prop osition 3.1 again to conclude that there is c ′′ > 0 suc h that g h N , 1 2 ( x ) ≳ xη h,c ′′ N for all x ∈ [0 , 1]. This, com bined with the inequality P [0 e φ N ≥ h ← → ∂ B N/ 2 ] ≥ E [ g h N , 1 2 ( ξ h N ) 1 0 <ξ h N ≤ 1 ] from ( 3.12 ), gives the desired low er b ound. W e complete the pro of of Prop osition 1.1 . □ No w, w e turn to the boundary one-arm probability . With the help the follo wing prop osition, w e can obtain Prop osition 1.2 in a similar fashion. W e thereby omit the pro of of Prop osition 1.2 . Prop osition 3.2. Ther e exist c, c ′ > 0 such that for any h ≥ 0 and x > 0 , (3.14) P h 0 e φ N ≥ h ← → ∂ i B N | e φ N , 0 = h + x p log N i ≤ cx (log N ) − 1 N − c ′ h 2 . Pr o of. Let ( e I t ) t ≥ 0 and P x b e the same as in the proof of Proposition 3.1 . Let V = ∂ i B N , and let M b e the exploration martingale M = M V . In con trast to the pro of of Prop osition 3.1 , w e adopt the non-normalized observ able and its corresponding martingale here to simplify computation. Let τ = inf { t ≥ 0 : e I t ∩ ∂ B N  = ∅} . By represen ting the quadratic v ariance of M in terms of Green’s functions (see, e.g., the pro of of [ 13 , Lemma 20]), there exist c 4 , c ′ 4 > 0 such that for all 0 ≤ s < t ≤ τ , (3.15) c 4 ( H N ( V , e I t ) − H N ( V , e I s )) ≤ ⟨ M ⟩ t − ⟨ M ⟩ s ≤ c ′ 4 ( H N ( V , e I t ) − H N ( V , e I s )) . where H N ( · ) was defined in ( 2.5 ). Note that by Lemma 2.3 , (3.16) H N ( V , e I 0 ) ≤ c 5 log N for some c 5 > 0. On the ev en t { τ < ∞} = { 0 e φ N ≥ h ← → ∂ B N } , b y first applying ( 3.15 ) then mo difying the pro of of [ 14 , (3.3)] to b ound H N ( V , e I τ ) from b elow, w e ha ve (3.17) ⟨ M ⟩ τ ≥ c 4 ( H N ( V , e I τ ) − H N ( V , e I 0 )) ≥ c 6 log N for some c 6 > 0. By the definition of the exploration pro cess, M t − M 0 ≥ h ( H N ( V , e I t ) − H N ( V , e I 0 )) − x p log N · H N ( V , e I 0 ) for all 0 ≤ t ≤ τ . Combining the last inequalit y with ( 3.15 ) and ( 3.16 ) giv es M t − M 0 ≥ ( c ′ 4 ) − 1 h ⟨ M ⟩ t − c 5 x √ log N One-arm probabilities for 2D GFF 12 for all 0 ≤ t ≤ τ . Thus, b y ( 3.17 ), P x [ τ < ∞ ] ≤ P x  M t − M 0 ≥ ( c ′ 4 ) − 1 h ⟨ M ⟩ t − c 5 x √ log N , ∀ t ∈ [0 , τ ]  ≤ P  B t ≥ ( c ′ 4 ) − 1 ht − c 5 x √ log N , ∀ t ∈ [0 , c 6 log N ]  , where in the second inequality we used Lemma 2.4 . The claim ( 3.14 ) then follows from another application of Lemma 2.1 and standard asymptotic analysis. □ 4. The Discrete Case In this section, we fo cus on the discrete GFF and give the pro of of Theorems 1.3 and 1.4 in Sections 4.1 and 4.2 , resp ectiv ely . 4.1. Discrete one-arm probabilit y . In this subsection, we pro v e Theorem 1.3 , whic h in spirit resem bles that of Proposition 1.1 except for an additional issue–in the discrete setup, the coun- terpart for the k ey estimate ( 3.4 ) do es not exactly hold true. More precisely , in contrast to the metric-gragh level set e E ≥ h N , when the exploration of E ≥ h N ends b efore reaching the target ∂ B N/ 2 , the b oundary v alues of the explored cluster are no longer exactly h , but almost surely smaller than h . F ortunately , leveraging the correlation of GFF, w e are able to show that for all large h , the harmonic av erage of those b oundary v alues is larger than h − c for some constant c with probabilit y sufficien tly close to 1, so that a w eak er v ersion of ( 3.4 ) still holds (see Proposition 4.2 for details). Throughout this subsection, w e set U = ∂ B ⌊ 3 4 N ⌋ , and let ( I t ) t ≥ 0 b e the exploration pro cess for E ≥ h N with source I 0 = { 0 } . T ake the exploration martingale M = M U . W e first in tro duce a technical lemma on the entropic repulsion of 2D GFF, which will b e needed b oth for this subsection (but only the sp ecial case when x ∈ Z 2 , δ = 1 and V = { x } ) and the pro of of Theorem 1.4 . Lemma 4.1. L et h ∈ R , x ∈ e B 3 4 N , y ∈ e Z 2 that satisfies | y − x | = δ ∈ (0 , 1] , and V b e any subset of e B N c ontaining x . Define the event (4.1) D : = { e φ N ,z ≥ h, ∀ z ∈ V } ∩ { e φ N ,z < h, ∀ z ∈ B N \ V , | z − x | ≥ 1 } . Then, ther e exists c 1 > 0 such that for al l lar ge N , (4.2) E [ e φ N ,y | D ] ≥ h − c 1 δ 1 2 , ∀ h ≤ p log N ; (4.3) E [ e φ N ,y | D ] ≥ 1 2 h, ∀ h > p log N . Pr o of. By [ 14 , Lemma 4.2], E [ e φ N ,y | D ] is greater than the one given e φ N ,z = h for all z ∈ V and the same condition at other sites. Therefore, w e will simply replace D in ( 4.1 ) by the following one D : = { e φ N ,z = h, ∀ z ∈ V } ∩ { e φ N ,z < h, ∀ z ∈ B N \ V , | z − x | ≥ 1 } , and prov e ( 4.2 ) and ( 4.3 ) in this case. Let x ∈ e B 3 4 N , y ∈ e Z 2 , and δ = | y − x | ∈ (0 , 1] b e as in the statement. W e ma y assume y / ∈ V , otherwise the estimates trivially hold. W e fo cus on the case h ≤ √ log N first and giv e a detailed pro of for ( 4.2 ). First, there is some c > 0 such that there exists a non-negative contin uous function g x satisfying g x ( x ) = 0, g x is harmonic on e Z 2 \{ x } and (4.4) sup z ∈ e Z 2   g x ( z ) − log  | z − x | + 2    < c. Let κ > 0 b e a parameter to b e determined later. Let f b e the function defined by (dep ending on x ) (4.5) f ( z ) = h − κg x ( z ) . Y. Bi, Y. Gao, X. Li 13 Then f is harmonic on e B N +1 \{ x } , and f ( z ) ≤ h for all z ∈ V . Moreo ver, since h ≤ √ log N , b y ( 4.4 ) and the fact that x ∈ e B 3 4 N , we can tak e N large enough such that for all z ∈ ∂ i B N , (4.6) f ( z ) ≤ p log N − κ  log  N / 4 + 2  − c  < 0 . Let ϕ b e another GFF on e B N +1 with the following conditions ϕ z = f ( z ) , ∀ z ∈ ∂ i B N ∪ V , and let D ′ b e the even t D ′ = { ϕ z < h, ∀ z ∈ B N \ V , | z − x | ≥ 1 } . Then, by [ 14 , Lemma 4.2], e φ N conditioned on D is s tochastically larger than ϕ conditioned on D ′ . In particular, for these tw o fields at the sp ecific p oin t y , we ha v e (4.7) E [ e φ N ,y |D ] ≥ E [ ϕ y |D ′ ] . F rom ab o v e, it remains to get a corresp onding lo wer b ound for E [ ϕ y |D ′ ]. First, by a union b ound and standard Gaussian estimates, w e can choose a large constan t κ (indep endent of δ and N ) in ( 4.5 ) such that for all large N , P [ D ′ ] > 1 2 . Therefore, ( E [ ϕ y |D ′ ] − E [ ϕ y ]) 2 ≤ 2( E [ ϕ y |D ′ ] − E [ ϕ y ]) 2 P [ D ′ ] ≤ 2V ar( ϕ y ) ≲ δ, whic h yields (4.8) E [ ϕ y |D ′ ] ≥ E [ ϕ y ] − cδ 1 2 for some c > 0. Moreov er, b y the facts that g x is locally Lipsc hitz con tinuous and that g x ( x ) = 0, w e hav e (4.9) E [ ϕ y ] = f ( y ) ≥ h − c ′ δ for some c ′ = c ′ ( κ ) > 0. Com bining ( 4.8 ) and ( 4.9 ), we obtain E [ ϕ y |D ′ ] ≥ E [ ϕ y ] − cδ 1 2 ≥ h − c 1 δ 1 2 for some c 1 = c 1 ( κ ) > 0, which further implies ( 4.2 ) by ( 4.7 ). T o obtain ( 4.3 ), we tak e κ = ch instead, where the constant c > 0 is chosen sufficiently small suc h that f ( y ) > 3 4 h , and then w e can pro ceed as ab o v e. W e omit the details. □ No w w e are ready to deduce the substitute for ( 3.4 ). This requires the use of Lemma 4.1 and Lemma 2.5 to con trol the mean and the fluctuation of the exploration martingale, resp ectiv ely . F or integers 1 ≤ k ≤ N , we in troduce the stopping times τ k b elo w (4.10) τ k : = inf { t ≥ 0 : I t ∩ ∂ B k  = ∅} . W e further record tw o sp ecial times which will be used frequently b elo w σ : = τ ⌊ N/ 2 ⌋ and σ 1 : = τ ⌊ √ N ⌋ . F or any x > 0, write as b efore (4.11) P x : = P [ · | φ N , 0 = h + x p log N ] , and denote the corresp onding exp ectation b y E x . W e ha ve the follo wing estimate. Prop osition 4.2. F or any h ∈ R , let ˆ h = h 4 (1 + 3 1 h ≤ √ log N ) . Define the event (4.12) H h N : = {∃ t ∈ [ σ 1 , σ ) such that M t < ( ˆ h − 3 c 1 ) H N ( U, I t ) } , wher e c 1 is fr om L emma 4.1 . Then, ther e is some c > 0 such that for al l x > 0 , (4.13) P x [ H h N ] ≲ ( N − 1 , h ≤ √ log N ; e − ch 2 , h > √ log N . One-arm probabilities for 2D GFF 14 As men tioned in the b eginning of this subsection, this prop osition will pla y the role of ( 3.4 ) in the discrete case, controlling the fluctuation of the exploration martingale around h during the time interv al [ σ 1 , σ ). Note that the lac k of con trol o v er [0 , σ 1 ) does not p ose an y issue, since ⟨ M ⟩ σ 1 ≲ (log N ) − 1 is sufficiently small (see the pro of of Prop osition 4.3 b elo w). Pr o of of Pr op osition 4.2 . T o obtain ( 4.13 ), w e control the exploration martingale ( M t ) t ∈ [ σ 1 ,σ ) along the in teger times and times b et w een them separately . More precisely , for an integer time k , consider the following t wo t ypes of even ts (4.14) E k : =  M k < ( h − 2 c 1 − 5 8 h 1 h> √ log N ) H N ( U, I k )  , and (4.15) F k : = ( inf t ∈ [ k,k +1] M t − M k < − ( c 1 + 1 8 h 1 h> √ log N ) · H N ( U, I k ) ) . Then, H h N implies that E k ∪ F k o ccurs for some integer k in [ σ 1 , σ ). Apparen tly , M j = M k for all j > k if I k = I k − 1 , and I k ⊋ I k − 1 if I k  = I k − 1 . In other w ords, the exploration pro cess is “strictly increasing” at eac h step b efore ev en tually b eing stopped. Therefore, there are at most O ( N 2 ) terms to be considered when getting ( 4.13 ) b y a union b ound. In other words, it suffices to show that there exists c > 0 such that for all x > 0 and k ≥ 1, (4.16) P x  { σ 1 ≤ k < σ } ∩ ( E k ∪ F k )  ≲ ( N − 3 , h ≤ √ log N ; e − ch 2 , h > √ log N . In the remainder of the pro of, we fix an in teger k and con trol E k and F k , separately . W e first deal with E k b y conditioning on any p ossible realization of I k . Let I b e an y connected set of v ertices such that 0 ∈ I ⊂ B N/ 2 . Let e I b e the metric graph asso ciated with I . Assume that P x [ { σ 1 ≤ k < σ } ∩ {I k = e I } ] > 0 (note that I k is generally a metric graph as we defined via linear in terpolation). It implies that 0 and ∂ B √ N are connected in I . Define the random v ector β I b y β I : = (1 { φ N ,x ≥ h } ) x ∈ I . The ev en ts {I k = e I } and { σ 1 ≤ k < σ } are b oth measurable with resp ect to β I . Let ω ∈ { 0 , 1 } I b e some realization of β I suc h that { β I = ω } ⊂ {I k = e I } , and define I + = { x ∈ I : ω x = 1 } , I − = { x ∈ I : ω x = 0 } . Note that 0 ∈ I + since x > 0. No w, w e estimate the exp ectation of M k giv en β I and F I + . Recalling the definition of D ( I ) in ( 2.7 ), we obtain that E [ M k | β I = w , F I + ] = X x ∈ D ( I ) H N ( U, x ; I ) E [ φ N ,x | β I = w , F I + ] (4.17) ≥ X x ∈ D ( I ) ∩ I + H N ( U, x ; I ) · h + X x ∈ D ( I ) ∩ I − H N ( U, x ; I ) inf z ∼ x E [ φ N ,x | φ N ,z ≥ h, φ N ,y < h, ∀ y ∈ B N , | y − z | ≥ 1] ≥ ( h − 1 2 h 1 h> √ log N − c 1 ) H N ( U, I ) , where we used Lemma 4.1 with V = { z } and δ = 1 in the last inequality . Next, w e derive an exp onen tial momen t b ound for M k (appropriately recentered and rescaled). Note that {I k = e I } is non-empt y , which indicates that for all x ∈ I − , there exists some x ′ ∈ I + suc h that x ∼ x ′ . Thus, b y Lemma 2.5 , for all r ∈ N , (4.18) µ (2 r ) [ M k | β I = w , F I + ] ≤ (2 r − 1)!! · 16 r ξ ( U, I ) r H N ( U, I ) 2 r , Y. Bi, Y. Gao, X. Li 15 where µ (2 r ) stands for the conditional (2 r )-th moment; see ( 2.8 ). Letting ζ k : = M k − E [ M k | β I , F I + ] ξ ( U, I ) 1 2 H N ( U, I ) , it follows from ( 4.18 ) that for an y 0 < b < 1 32 , (4.19) E [ e bζ 2 k | β I = w , F I + ] = ∞ X r =0 b r r ! ξ ( U, I ) r H N ( U, I ) 2 r µ (2 r ) [ M k | β I = w , F I + ] ≤ ∞ X r =0 (16 b ) r r ! (2 r − 1)!! ≤ ∞ X r =0 (32 b ) r < ∞ . F urthermore, we need to giv e an upp er b ound for ξ ( U, I ) (recall its definition in ( 2.10 )). W rite I ′ = I \ D ( I ). W e hav e for all y ∈ U and x ∈ I ′ , H N ( y , x ; I ′ ) ≤ H N ( y , x ; I ′ ) G ( e B N +1 \ I ′ ) ∪{ x } ( x, x ) = G ( e B N +1 \ I ′ ) ∪{ x } ( y , x ) = H N ( x, y ; ( I ′ \ { x } ) ∪ { y } ) G ( e B N +1 \ I ′ ) ∪{ x } ( y , y ) . Since G ( e B N +1 \ I ′ ) ∪{ x } ( y , y ) ≤ G e B N +1 ( y , y ) ≲ log N , and b y Lemma 2.2 (note that I ′ has diameter at least 1 2 √ N ), for some a > 0, H N ( x, y ; ( I ′ \ { x } ) ∪ { y } ) ≤ P x  τ I ′ \{ x } > τ x + ∂ e B √ N / 4  ≲ N − 4 a . Hence, we ha v e H N ( U, x ; I ′ ) ≲ log N · N − 4 a ≲ N − 2 a . Moreo v er, by Lemma 2.3 , H N ( U, I ) ≳ (log N ) − 1 . Therefore, (4.20) ξ ( U, I ) = sup x ∈ I ′ H N ( U, x ; I ′ ) /H N ( U, I ) ≲ N − a . When h ≤ √ log N , by ( 4.17 ), ( 4.19 ) and ( 4.20 ), using Cheb yshev’s inequality , w e get P [ M k < ( h − 2 c 1 ) H N ( U, I ) | β I = w , F I + ] ≤ P [ | ζ k | ≥ c 1 ξ ( U, I ) − 1 2 ] ≤ exp  − bc 2 1 ξ ( U, I )  E [ e bζ 2 k | β I = w , F I + ] ≲ e − cN a ≲ N − 3 . (4.21) Similarly , when h > √ log N , for some c > 0, (4.22) P [ M k < ( 3 8 h − 2 c 1 ) H N ( U, I ) | β I = w , F I + ] ≤ P [ | ζ k | ≥ ( 1 8 h + c 1 ) ξ ( U, I ) − 1 2 ] ≲ e − ch 2 . Com bined, we obtain the following desired upp er b ound for E k , (4.23) P x [ { σ 1 ≤ k < σ } ∩ E k ] = E x  P x [ M k < ( h − 2 c 1 − 5 8 h 1 h> √ log N ) H N ( U, I k ) | I k ]1 { σ 1 ≤ k<σ }  ≲ ( N − 3 , h ≤ √ log N ; e − ch 2 , h > √ log N . Next, w e turn to the ev en t F k . First, analogous to ( 3.8 ), we hav e the follo wing standard estimate (4.24) c 2 ( H N ( U, I t ) − H N ( U, I s )) ≤ ⟨ M ⟩ t − ⟨ M ⟩ s ≤ c 3 ( H N ( U, I t ) − H N ( U, I s )) for some c 2 , c 3 > 0 and all t > s ≥ 0. F or σ 1 ≤ k < σ , the ab o v e inequality and another application of Lemma 2.2 give ⟨ M ⟩ k +1 − ⟨ M ⟩ k ≲ H N ( U, I k +1 ) − H N ( U, I k ) ≤ sup x ∈I k +1 \I k H N ( x, ∂ B N ; ∂ B N ∪ I k ) ≲ N − a , One-arm probabilities for 2D GFF 16 where H N ( x, ∂ B N ; ∂ B N ∪ I k ) : = P y ∈ ∂ B N H N ( x, y ; ∂ B N ∪ I k ). That is, there exists c 4 > 0 suc h that ⟨ M ⟩ k +1 − ⟨ M ⟩ k ≤ c 4 N − a . Also, note that by Lemma 2.3 , there exists c 5 > 0 suc h that H N ( U, I k ) ≥ c 5 (log N ) − 1 . Recalling F k from ( 4.15 ) and using Lemma 2.4 , when h ≤ √ log N , (4.25) P x [ { σ 1 ≤ k < σ } ∩ F k ] ≤ P " inf t ∈ [0 ,c 4 N − a ] B t < − c 1 c 5 (log N ) − 1 # ≲ N − 3 . Similarly , there is some c > 0 suc h that when h > √ log N , (4.26) P x [ { σ 1 ≤ k < σ } ∩ F k ] ≤ P " inf t ∈ [0 ,c 4 N − a ] B t < − ( 1 8 h + c 1 ) c 5 (log N ) − 1 # ≲ e − ch 2 . The conclusion ( 4.16 ) (and hence ( 4.13 )) then follo ws from ( 4.23 ), ( 4.25 ) and ( 4.26 ). This finishes the pro of. □ T o combine the pro of of Theorem 1.3 and ( 1.11 ) in the same framework, we derive the following upp er b ound on the probability of a slightly general v ersion of one-arm even t. Prop osition 4.3. Ther e is a c onstant ρ > 0 such that for any inte ger k ∈ [ N (log N ) − 1 , N / 2] and any h ≤ √ log N , we have (4.27) P [0 φ N ≥ h ← → ∂ B k ] ≲ e η h,ρ N ,k , wher e e η h,ρ N ,k : =    | h |∨ √ log N − log k √ log N , h ≤ 0; q log N − log k log N exp  − ρh 2 log N − log k  , h > 0 . Pr o of. As in the pro of of Prop osition 1.1 , it suffices to deal with the case h ≥ − √ log N . F ur- thermore, we ma y assume that N is larger than an y fixed constan t. Indeed, arbitrarily fix c > 0, then for any N < c and h ∈ R , there exists c ′ > 0 such that w e hav e the following estimate: P  0 φ N ≥ h ← → ∂ B k  ≤ P [ φ N , 0 ≥ h ] ≲ ( 1 , h ≤ 0; e − c ′ h 2 , h > 0 . As b efore, the k ey argument is to estimate the asso ciated exploration martingale. Recall τ k from ( 4.10 ). By ( 4.24 ) and Lemma 2.3 , there are c 6 , c 7 > 0 suc h that on the ev en t { 0 φ N ≥ h ← → ∂ B k } , (4.28) ⟨ M ⟩ τ k ≥ c 2 ( H N ( U, I τ k ) − H N ( U, I 0 )) ≥ c 6 (log N − log k ) − 1 , ⟨ M ⟩ σ 1 ≤ c 3 H N ( U, B √ N ) ≤ c 7 log N . Let H h N b e the ev en t in ( 4.12 ), and recall ξ h N in ( 3.11 ). Using ( 4.24 ) and Lemma 2.3 again, there exists c 8 > 0 such that on the even t { τ k < ∞} ∩ ( H h N ) c , for all t ∈ [ σ 1 , τ k ), (4.29) M t − M 0 ≥ ( h − 3 c 1 ) H N ( U, I t ) − ( h + ξ h N p log N ) H N ( U, I 0 ) = ( h − 3 c 1 )( H N ( U, I t ) − H N ( U, I 0 )) − (3 c 1 + ξ h N p log N ) H N ( U, I 0 ) ≥ c 8 ( h − 3 c 1 ) ⟨ M ⟩ t − c 8 (3 c 1 + ξ h N p log N )(log N ) − 1 . Considering the even t (4.30) A k : = n M t − M 0 ≥ c 8 ( h − 3 c 1 ) ⟨ M ⟩ t − c 8 (3 c 1 + ξ h N √ log N ) log N , ∀ t satisfying c 7 (log N ) − 1 < ⟨ M ⟩ t < c 6 (log N − log k ) − 1 o , then ( 4.28 ) and ( 4.29 ) together imply the following inclusion (4.31) { 0 φ N ≥ h ← → ∂ B k } ⊂ H h N ∪ ( { τ k < ∞} ∩ ( H h N ) c ) ⊂ H h N ∪ A k . Y. Bi, Y. Gao, X. Li 17 By Prop osition 4.2 , P x [ H h N ] ≲ N − 1 , where P x is defined by ( 4.11 ). It follo ws from ( 4.31 ) that (4.32) P x [0 φ N ≥ h ← → ∂ B k ] ≤ P x [ H h N ] + P x [ A k ] ≲ N − 1 + P x [ A k ] . Note that P x = P [ · | ξ h N = x ] b y our notation. By Lemmas 2.1 , 2.4 and asymptotic estimates ( 2.3 ), ( 2.4 ), there is c > 0 such that P x [ A k ] = P h B t − B c 7 log N ≥ c 8 ( h − 3 c 1 )( t − c 7 log N ) − g ( x ) √ log N − B c 7 log N , ∀ t ∈  c 7 log N , c 6 log N − log k i ≲ E    g ( x ) + p log N · B c 7 log N  e η h,c N ,k 1 0 ≤ g ( x )+ √ log N · B c 7 log N ≤ q log N log N − log k   + P   g ( x ) + p log N · B c 7 log N ≥ s log N log N − log k   , where g ( x ) : = c 8 ( − c 7 h +3 c 1 c 7 +3 c 1 + x √ log N ) √ log N . Let ξ and ζ b e indep enden t copies of g ( ξ h N ) and √ log N · B c 7 log N , resp ectiv ely . Then, by ( 4.32 ) and the ab o v e b ound for P x [ A k ], P [0 φ N ≥ h ← → ∂ B k ] ≲ N − 1 + P [ A k ] ≲ N − 1 + e η h,c N ,k E  ( ξ + ζ ) 1 0 ≤ ξ + ζ ≤ q log N log N − log k  + P   ξ + ζ ≥ s log N log N − log k   ≲ N − 1 + e η h,c N ,k + exp  − c ′ log N log N − log k  , where the last inequalit y holds since ξ + ζ is Gaussian with mean and v ariance of constan t order (note that E [ ξ h N ] ∈ [ − 1 , 1] since | h | ≤ √ log N ). Note that for h ≤ 0, η h,c N is the dominating term on the right-hand side ab o v e; while for 0 < h ≤ √ log N , w e ma y c ho ose ρ > 0 small enough so that max ( N − 1 , e η h,c N ,k , exp  − c ′ log N log N − log k  ) ≤ e η h,ρ N ,k . The prop osition then follows. □ No w, we use Prop osition 4.3 to prov e Theorem 1.3 . Pr o of of The or em 1.3 . T o obtain the low er b ounds in ( 1.6 ) and ( 1.7 ), just note that P [0 φ N ≥ h ← → ∂ B N/ 2 ] ≥ P [0 e φ N ≥ h ← → ∂ B N/ 2 ] , and apply Prop osition 1.1 . No w we turn to upp er b ounds. W e first deal with case h ≤ √ log N . By Proposition 4.3 , there exists some ρ > 0 such that P [0 φ N ≥ h ← → ∂ B N/ 2 ] ≲ e η h,ρ N ,N/ 2 ≲ η h,ρ (log 2) − 1 N , where the function η h,ρ N is given by ( 3.13 ). This gives the desired b ounds. When h > √ log N , w e can pro ceed as ab o v e with A N/ 2 (defined in ( 4.30 )) replaced by the follo wing even t ˆ A N/ 2 : = n M t − M 0 ≥ c 8 ( 1 2 h − 3 c 1 ) ⟨ M ⟩ t − c 8 (3 c 1 + 1 2 h + ξ h N √ log N ) log N , ∀ t satisfying c 7 (log N ) − 1 < ⟨ M ⟩ t < c 6 (log N − log k ) − 1 o , One-arm probabilities for 2D GFF 18 and apply the follo wing inequality , analogous to ( 4.32 ), P [0 φ N ≥ h ← → ∂ B N/ 2 ] ≤ P [ H h N ] + P [ ˆ A N/ 2 ] . In fact, by Prop osition 4.2 , P [ H h N ] ≲ e − ch 2 for some c > 0. Also, for some c ′ , c ′′ > 0, P [ ˆ A N/ 2 ] ≤ P h B c 6 log 2 ≥ c 8 ( 1 2 h − 3 c 1 ) c 6 log 2 − c 8 (3 c 1 + 1 2 h + ξ h N √ log N ) log N i ≤ P h B c 6 log 2 ≥ c ′ h − c 8 ξ h N √ log N i ≲ e − c ′′ h 2 . (4.33) This yields the desired upp er b ounds on P [0 φ N ≥ h ← → ∂ B N/ 2 ]. □ Sk etc h of pro of of ( 1.11 ) . This estimate on the chemical distance on E ≥ h N can b e obtained b y adapting the argumen t from [ 13 ], the corresponding result for e E ≥ h N . All ingredients therein can b e easily adapted with the metric-graph GFF replaced by the corresponding arguments for discrete GFF, except for [ 13 , Lemma 24], whose coun terpart for the discrete GFF is stated and pro v ed b elow. Lemma 4.4. F or any h ∈ R and 0 < α < β < γ < 1 , ther e exists some c = c ( h, α, β , γ ) > 0 such that for al l j ∈ N , k ≥ k 0 : = ( β − α ) N/ √ log N and v ∈ ∂ B αN + k , (4.34) P " v φ N ≥ h ← → B αN      φ N ,v = h + 2 j p log N # ≤ c 2 j √ log N p log N − log k . The lemma is pro ved using the same technique as in the pro of of Theorem 1.3 . Pr o of. W rite P ′ for the ab o v e conditional law. In this pro of, we fix h ∈ R , and the constan ts in tro duced later may implicitly dep end on parameters α , β , γ and h . Consider ( I ′ t ) t ≥ 0 , the exploration of E ≥ h N with source I ′ 0 = { v } . Let k ′ = k ∧ ( 1 − γ 2 N ), U ′ = ∂ B ( v, 3 4 (1 − γ ) N ) and M ′ = M U ′ . Define the ev en t H ′ : = n ∃ t ∈ [ σ ′ 1 , σ ′ ) s.t. M ′ t < ( h − 3 c 1 ) H N ( U ′ , I ′ t ) o , where σ ′ = inf { t ≥ 0 : I ′ t ∩ ∂ B ( v , k ′ )  = ∅} , σ ′ 1 = inf { t ≥ 0 : I ′ t ∩ ∂ B ( v , √ N )  = ∅} . Then, similar to Prop osition 4.2 (when h ≤ √ log N ), we conclude P ′ [ H ′ ] ≲ N − 1 . Thus, parallel to ( 4.32 ), P ′  v φ N ≥ h ← → B αN  ≤ P ′  v φ N ≥ h ← → ∂ B ( v , k ′ )  ≤ P ′ [ H ′ ] + P ′ [ A ′ k ] . Here, A ′ k : = n M ′ t − M ′ 0 ≥ c ⟨ M ′ ⟩ t − c ′ log N − c ′′ 2 j √ log N , ∀ t ≥ 0 suc h that c − (log N ) − 1 < ⟨ M ′ ⟩ t < c + (log N − log k ) − 1 o , (4.35) where c, c ′ , c ′′ , c − , c + > 0 are appropriately chosen parameter-dep enden t constants. Applying Lemmas 2.1 and 2.4 , we obtain the desired estimate. □ 4.2. Comparison with the metric-graph case: proof of Theorem 1.4 . T o establish the discrepancy of the one-arm probabilities, we sho w that given the exploration of e E ≥ h N from 0, with the restriction that it stops within the ann ulus e A rN ,N / 4 (whic h happens with probabilit y of order (log N ) − 1 / 2 ), the conditional probability that the corresp onding discrete exploration of E ≥ h N reac hing ∂ B N/ 2 is b ounded aw ay from 0. Let C 0 b e the cluster in e E ≥ h N con taining 0 with the con v en tion that C 0 = ∅ if e φ N , 0 < h . Let C 0 such that for al l N , (4.38) P [Λ] ≥ c √ log N . Pr o of. Recall the definition of the even t C h N ,k,l from ab o v e Prop osition 2.7 . It suffices to give a lo w er b ound on the probability of the even t Λ ′ : = C h N , 3 4 N ,N ∩ C h N ,rN , 2 r N ∩ ( 0 e φ N ≥ h ← → ∂ e B 2 rN , 0 e φ N ≥ h  ← → ∂ e B N/ 4 ) , since apparently , Λ ′ ⊂ Λ. By Prop osition 3.1 and an integration similar to the pro of of Prop o- sition 1.1 , there exist c ( h ) , c ′ ( h ) > 0 such that c s | log r | log N < P [0 e φ N ≥ h ← → ∂ B 2 rN ] < c ′ s | log r | log N . By Prop osition 2.7 , there exists c ′′ = c ′′ ( r , h ) > 0 such that min  P [ C h N , 3 4 N ,N ] , P [ C h N ,rN , 2 r N ]  > c ′′ . Th us, by FK G inequality , one may c hoose r ∈ (0 , 1 4 ) sufficiently small suc h that P [Λ ′ ] ≥ P [ C h N , 3 4 N ,N ] · P [ C h N ,rN , 2 r N ] · P [0 e φ N ≥ h ← → ∂ B 2 rN ] − P [0 e φ N ≥ h ← → ∂ B N/ 2 ] ≥  ( c ′′ ) 2 c p | log r | − c ′ p log 4  · (log N ) − 1 / 2 ≳ 1 √ log N . The conclusion then follo ws since Λ ′ ⊂ Λ. □ F rom now on till the end of this section, w e c ho ose some r (in the definition of Λ = Λ N ,h,r in ( 4.37 )) suc h that ( 4.38 ) holds. T o ligh ten notation, we use P ∗ to denote the conditional probabilit y given all the information of GFF at C 0 ∪ C 0 such that (4.41) P ∗ [Γ φ N ≥ h ← → ∂ B N/ 2 ] ≥ c. One-arm probabilities for 2D GFF 20 Pr o of. The pro of is based on the same ideas presen ted in [ 14 , Section 4]. W e assume that Λ holds b elo w, which is measurable with resp ect to C 0 ∪ C 0, such that (4.42) P ∗ [ F | ˜ E ] = 1 − o N (1) , where F : = {∃ 0 < t < σ ′ s.t. M t < h − ∆ · H ∗ N ( ∂ B 5 8 N , I t ) } . Note that Λ en tails Γ ⊂ B N/ 2 and C 0 such that for all 0 < t < σ ′ , H ∗ N ( ∂ B 5 8 N , I t ) =  H ∗ N ( ∂ B 5 8 N , I t ) − H ∗ N ( ∂ B 5 8 N , Γ)  + 1 2 H ∗ N ( ∂ B 5 8 N , Γ) + 1 2 H ∗ N ( ∂ B 5 8 N , Γ) ≥ c ( ⟨ M ⟩ t + V ar ∗ ( M 0 ) + 1) , further yielding F ⊂ {∃ 0 < t < σ ′ s.t. M t − h < − c ∆( ⟨ M ⟩ t + V ar ∗ ( M 0 ) + 1) } . Applying Lemma 2.4 again, we may view M t − M 0 as a time-c hanged standard Brownian motion. F urther noting that M 0 − h is a ce n tered Gaussian v ariable with v ariance V ar ∗ ( M 0 ), we ma y view the pro cess M t − h = ( M t − M 0 ) + ( M 0 − h ) as the part of a standard Bro wnian motion after time V ar ∗ ( M 0 ). Hence, we obtain P ∗ [ F ] ≤ P [ ∃ t ≥ 0 s.t. B t ≤ − c ∆( t + 1)] = : 1 − p 0 , where p 0 > 0 by Lemma 2.1 (sending T → ∞ ). This combined with ( 4.42 ) gives that for large N , P ∗ [ ˜ E ] ≤ P ∗ [ F ] P ∗ [ F | ˜ E ] < 1 − 1 2 p 0 . This implies ( 4.41 ) immediately . W e next turn to the pro of of ( 4.42 ). Note that it suffices to b ound from b elo w the probability of F giv en {I ∞ = e I } , where I is any connected set of v ertices satisfying Γ ⊂ I ⊂ B N/ 2 \ C 0 and e I is the metric graph asso ciated with I . Moreo v er, {I ∞ = e I } is measurable with resp ect to β I : = ( 1 { φ N ,x ≥ h } ) x ∈ I , then it suffices to show on {I ∞ = I } , (4.43) P ∗ [ M ∞ < ( h − ∆) H ∗ N ( ∂ B 5 8 N , I ) | β I ] = 1 − o N (1) , a . s . The strategy for the pro of of ( 4.43 ) giv en b elo w is in fact very similar to ( 4.16 ) (but note that the desired inequality is in a rev erse direction). Consider any b I ∈ { 0 , 1 } I suc h that { β I = b I } ⊂ {I ∞ = I } . Let I + = { x ∈ I : b I ,x = 1 } , I − = I \ I + , and recall the definition of D ( I ) from ( 2.7 ). Note that we hav e D ( I ) ⊂ I − . Moreo v er, on the even t { β I = b I } , under P ∗ , (4.44) M ∞ = X x ∈ D ( I ) H ∗ N ( ∂ B 5 8 N , x ; I ) φ N ,x . Y. Bi, Y. Gao, X. Li 21 By [ 14 , Lemma 4.2] (comparison of GFFs with differen t conditions), w e get that for all x ∈ D ( I ), E ∗ [ φ N ,x | β I = b I , F I + ] ≤ E ∗ [ φ N ,x | φ N ,x < h, β I + = b I + , F I + ] ≤ E ∗ [ φ N ,x ∧ h | β I + = b I + , F I + ] ≤ h − ce − c ′ m 2 x , (4.45) where constants c, c ′ > 0 and m x : = E ∗ [ φ N ,x | β I + = b I + , F I + ] − h > 0 . W e claim that Y ′ : = P x ∈ D ( I ) H ∗ N ( ∂ B 5 8 N , x ; I ) m x satisfies (4.46) P ∗ [ Y ′ ≤ c ′′ H ∗ N ( ∂ B 5 8 N , I ) | β I = b I ] = 1 − o N (1) for some constant c ′′ > 0. W e now prov e ( 4.43 ) assuming ( 4.46 ) for the time b eing. By Mark o v’s inequality , Y ′ ≤ c ′′ H ∗ N ( ∂ B 5 8 N , I ) implies (4.47) X x ∈ D ( I ) H ∗ N ( ∂ B 5 8 N , x ; I ) 1 m x ≤ 2 c ′′ ≥ 1 2 H ∗ N ( ∂ B 5 8 N , I ) . Com bining ( 4.44 ), ( 4.45 ) and ( 4.47 ), we obtain that with probability 1 − o N (1), E ∗ [ M ∞ | β I = b I , F I + ] = X x ∈ D ( I ) H ∗ N ( ∂ B 5 8 N , x ; I ) E ∗ [ φ N ,x | β I = b I , F I + ] ≤ hH ∗ N ( ∂ B 5 8 N , I ) − X x ∈ D ( I ) ce − c ′ m 2 x H ∗ N ( ∂ B 5 8 N , x ; I ) ≤ hH ∗ N ( ∂ B 5 8 N , I ) − ce − c ′ (2 c ′′ ) 2 X x ∈ D ( I ) H ∗ N ( ∂ B 5 8 N , x ; I ) 1 m x ≤ 2 c ′ ≤ ( h − 2∆) H ∗ N ( ∂ B 5 8 N , I ) , where we set ∆ = 1 4 ce − c ′ (2 c ′′ ) 2 . Moreo ver, it is not hard to c heck that both ( 4.18 ) and ( 4.20 ) hold under the measure P ∗ for I in the curren t context, whic h further imply that (4.48) V ar ∗ ( M ∞ | β I = b I , F I + ) = o N (1) H ∗ N ( ∂ B 5 8 N , I ) 2 . It follows from Cheb yshev’s inequality that P ∗ [ M ∞ > ( h − ∆) H ∗ N ( ∂ B 5 8 N , I ) | β I ] ≤ P ∗ [ M ∞ − E ∗ [ M ∞ | β I ] > ∆ H ∗ N ( ∂ B 5 8 N , I ) | β I ] ≤ o N (1) H ∗ N ( ∂ B 5 8 N , I ) 2 (∆ H ∗ N ( ∂ B 5 8 N , I )) 2 = o N (1) , finishing the pro of of ( 4.43 ) and hence of ( 4.41 ). It remains to sho w ( 4.46 ). Let D ′ = { x : H ∗ N ( ∂ B 5 8 N , x ; I + ) > 0 } and note that Y ′ = X x ∈ D ′ H ∗ N ( ∂ B 5 8 N , x ; I + )( φ N ,x − h ) . The desired b ound on Y ′ then follows by adapting the pro of of ( 4.43 ). First, observ e that ev ery v ertex in D ′ has a nearest neighbor in I − , so an adaption of Lemma 4.1 yields that for some c > 0, E ∗ [ Y ′ | β I = b I , F I − ] ≤ cH ∗ N ( ∂ B 5 8 N , I + ) ≤ cH ∗ N ( ∂ B 5 8 N , I ) . Second, similar to ( 4.48 ), we ha v e V ar ∗ ( Y ′ | β I = b I , F I − ) ≤ o N (1) H ∗ N ( ∂ B 5 8 N , I ) 2 . W e then conclude by applying Cheb yshev’s inequality again. F or the case h < 0, we consider a new field e φ − N : = e φ N − f h , where f h ( x ) = | h | H ∗ N ( x, ∂ B N +1 ) for all x ∈ e B N +1 \ ( C 0 ∪ C 0. This follows from the same argument as ab o v e. W e conclude the pro of of Prop osition 4.6 . □ W e finally turn to Theorem 1.4 . Proposition 4.6 implies that with uniformly p ositive P ∗ - probabilit y (recall that P ∗ is defined in ( 4.39 )), there exists a “piv otal v ertex” for the even t { 0 φ N ≥ h ← → ∂ B N/ 2 } . Thus, it remains to pro v e that the cost of “op ening” that piv otal vertex is a constan t (c.f. ( 4.49 ) b elo w). Pr o of of The or em 1.4 . Recall C ∗ 0 and Γ in ( 4.36 ) and ( 4.40 ) resp ectiv ely , and further define Γ ′ : = { x ∈ Z 2 : 0 < dist( x, C ∗ 0 ) < 1 } . Note that on { Γ φ N ≥ h ← → ∂ B N/ 2 } , we may c hoose some random v ertex v ∈ Γ ′ suc h that 0 is connected to ∂ B N/ 2 in E ≥ h N whenev er φ N ,v ≥ h (w e pick one deterministically if there are m ultiple candidates). Let β ∗ : = ( 1 φ N ,x ≥ h ) x ∈ B N \ ( C 0 ∪C 0 suc h that (4.49) P ∗ [ φ N ,v ≥ h | β ∗ ] ≥ c. Assuming ( 4.49 ), one has, P ∗ [0 φ N ≥ h ← → ∂ B N/ 2 ] ≥ P ∗ [Γ φ N ≥ h ← → ∂ B N/ 2 , φ N ,v ≥ h ] = E ∗ " 1 { Γ φ N ≥ h ← → ∂ B N/ 2 } P ∗ [ φ N ,v ≥ h | β ∗ ] # ≥ c, where the last step follows from ( 4.49 ) and Prop osition 4.6 . Hence, P  0 φ N ≥ h ← → ∂ B N/ 2   Λ  ≥ c , whic h, combined with Prop osition 4.5 , yields P  0 φ N ≥ h ← → ∂ B N/ 2  − P  0 e φ N ≥ h ← → ∂ B N/ 2  ≥ P [Λ] P  0 φ N ≥ h ← → ∂ B N/ 2   Λ  ≥ c √ log N for some c = c ( h ) > 0, proving the theorem. W e now pro v e ( 4.49 ). Let δ = dist( v, C ∗ 0 ). By the Marko v prop erty of the GFF, we ha v e the orthogonal decomposition φ N ,v = Y + Z , where Y is a c en tered Gaussian v ariable independent of F B N \ ( C 0 ∪C 0, and Z = E ∗ [ φ N ,v | F B N \ ( C 0 ∪C

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