On the critical fugacity of the hard-core model on regular bipartite graphs

On the critical fugacit y of the hard-core mo del on regular bipartite graphs Daniel Hadas ∗ Ron P eled † Marc h 31, 2026 Abstract W e establish long-range order for the hard-core mo del on a ﬁnite, regu- lar bipartite graph ab o v e a threshold fugacity giv en in terms of expansion parameters of the graph. The result applies to the d -dimensional h yp er- cub e graph and, more generally , to d -dimensional discrete tori of ﬁxed side length, pro ving long-range order at fugacities λ ≥ Ω( log d d ) . F urther- more, we use reﬂection p ositivit y to transfer the result to the lattice Z d , v erifying the long-standing b elief that its critical fugacity is of the form d − 1+ o (1) as d → ∞ . 1 In tro duction 1.1 The hard-core mo del Giv en a ﬁnite graph G = ( V , E ) and parameter λ > 0 , the har d-c or e mo del on G at fugacity λ is the probability measure µ G on indep enden t sets of G which assigns each indep enden t set probability prop ortional to λ raised to the size of the set. F ormally , we identify subsets of G with their characteristic functions, writing Ω G hc : = { σ ∈ { 0 , 1 } V : uv ∈ E = ⇒ σ u σ v = 0 } (1) for the family of hard-core conﬁgurations. F or σ ∈ { 0 , 1 } V , we write | σ | := P v ∈ V σ v , and deﬁne a measure ζ G on Ω G hc b y ζ G ( σ ) := λ | σ | . (2) The hard-core measure on G is giv en b y normalizing ζ G to a probabilit y measure: µ G ( σ ) = 1 Z G ζ G ( σ ) (3) ∗ School of Mathematical Sciences, T el A viv Universit y danielhadas1@mail.tau.ac.il † Department of Mathematics, Universit y of Maryland, College Park peledron@umd.edu 1 where Z G : = ζ G (Ω G hc ) is called the p artition function. F or brevity , w e ha v e omitted the λ dep endence from the notation. Do typical conﬁgurations of the hard-core model exhibit some kind of long- range order? Conv ersely , do the correlations in σ decay at long distances? At lo w fugacities, the sampled independent sets tend to be sparse, with σ u , σ v b eing nearly uncorrelated for vertices u, v at large graph distance. The b est general result of this kind is b y W eitz, who shows in [W ei06, Corollary 2.6] that the hard-core mo del on G exhibits str ong sp atial mixing when λ ≤ λ ∆ := (∆ − 1) ∆ − 1 (∆ − 2) ∆ , (4) with ∆ denoting the maximal degree in G . Here, strong spatial mixing means that for ev ery subset Λ ⊂ V , vertex v , and τ 1 , τ 2 ∈ Ω G hc , we hav e that | µ G ( σ v = 1 | σ Λ = τ 1 ) − µ G ( σ v = 1 | σ Λ = τ 2 ) | decays to zero with the distance of v from the set { u : τ 1 u  = τ 2 u } , at a rate which dep ends only on ∆ and λ . Here and later, we write σ Λ for the restriction of σ to Λ . The b ound (4) is sharp, as it is kno wn that such mixing fails for the hard-core mo del on (ﬁnite subsets of ) the ∆ -regular tree at any higher fugacity . W e note that λ ∆ ∼ e ∆ as ∆ → ∞ . (5) Alternativ e conditions implying the deca y of correlations follo w from Dobrushin’s uniqueness condition [Dob68a], or from v an den Berg–Maes’ disagreemen t p er- colation [BM94], but in terms of their dep endence on the maximal degree ∆ , these apply in parameter ranges smaller than (4) (see discussion in [PS20, Sec- tion 3.1.1]). A t high fugacities, dep ending on the graph structure, man y forms of long range order ma y emerge. These can serv e as models for v arious states of matter suc h as crystals and liquid crystals (see discussions in [HJ24], [DG13], [HP25], [P el22]). In this pap er, w e fo cus on the sp eciﬁc case that the graph G is simple, connected, regular and bipartite, with bipartition ( E , O ) . In this case, the largest indep enden t sets are E and O and one ma y exp ect that typical indep enden t sets sampled from µ G will b e mostly con tained in one of these. Our goal is to obtain new quantitativ e b ounds on the minimal fugacity λ at which such long-range order arises. The next sections detail our results: ﬁrst for the lattice Z d , then for discrete tori graphs and ﬁnally for regular bipartite graphs with suitable expansion prop erties. W e chose to b egin with the lattice result as it is simpler to state, but we p oin t out that its pro of uses the result for ﬁnite graphs as its main ingredient. 1.2 The hard-core mo del on Z d In the context of statistical physics, it is natural to study the hard-core mo del on the lattice Z d . This is the graph whose vertex set, also denoted Z d , is the 2 set of d -dimensional v ectors with in teger co ordinates, with tw o vectors adjacent if their Euclidean distance is 1. On such an inﬁnite graph, the replacement for the probability measure (3) is the notion of Gibbs measures. A Gibbs me asur e for the hard-core mo del on Z d at fugacit y λ is an y probabilit y measure µ λ on Ω Z d hc with the following prop ert y: Supp ose σ is sampled from µ λ . F or any ﬁnite Λ ⊂ Z d , conditioned on the restriction σ Λ c , the probability of each σ is prop ortional to λ | σ Λ | . The fundamental problem is then to decide, for eac h fugacity λ , whether the hard-core mo del admits a unique Gibbs measure or multiple ones, with m ulti- plicit y indicating long-range order (in this w ay , the inﬁnite graph setting is more elegan t than the ﬁnite graph s etting, as it has a canonical notion of long-range order). W e also mention that, using standard monotonicity (FK G) prop erties, there are multiple Gibbs measures if and only if the tw o Gibbs measures ob- tained as inﬁnite-volume limits with ev en/o dd b oundary conditions are distinct (see [vS94, Lemma 3.2]). These latter measures are extremal and in v ariant to the parity-preserving automorphisms of Z d . One thus deﬁnes λ − c ( d ) := inf { λ : the hard-core mo del on Z d has multiple Gibbs measures at fugacity λ } , λ + c ( d ) := sup { λ : the hard-core mo del on Z d has a unique Gibbs measure at fugacity λ } . Dobrushin [Dob68a, Dob68b] prov ed that the hard-core mo del on Z d has a phase tr ansition for eac h d ≥ 2 in the sense that 0 < λ − c ( d ) ≤ λ + c ( d ) < ∞ . While it is exp ected that λ − c ( d ) = λ + c ( d ) , this remains unkno wn (and there are examples of other inﬁnite graphs for which λ − c < λ + c , i.e., multiple transitions o ccur [BHW99]). A question of enduring interest is to ﬁnd the order of magnitudes of λ − c ( d ) and λ + c ( d ) in the limit d → ∞ . F rom one side, the general b ound (4) applies, and sho ws that (also noting (5)) λ − c ( d ) ≥ λ 2 d ∼ e 2 d as d → ∞ . (6) F rom the other side, while Dobrushin’s result only giv es an exp onen tially gro w- ing b ound on λ + c ( d ) , a breakthrough result of Galvin–Kahn [GK04] from 2004 sho wed that λ + c ( d ) → 0 as d → ∞ , obtaining the b ound λ + c ( d ) ≤ C log 3 / 4 d d 1 / 4 (7) for an absolute constant C > 0 . It was also said in [GK04] that natural guesses for the correct order of magnitude of λ + c ( d ) are log d d or 1 d . The current state- of-the-art, and the only improv emen t so far to (7), is by Samotij–Peled [PS14] who show ed that λ + c ( d ) ≤ C log 2 d d 1 / 3 . 3 Our main result on Z d , stated next, ﬁnally closes most of the gap to the low er b ound (6), showing that λ + c ( d ) = d − 1+ o (1) as d → ∞ . Theorem 1.1. Ther e exists C > 0 such that the har d-c or e mo del on Z d admits multiple Gibbs me asur es at e ach fugacity λ > C log d d in dimensions d ≥ 2 . As a byproduct of our analysis we also get a b ound on the sp eciﬁc free energy of the mo del, see Corollary 1.4 b elo w. W e conjecture that the low er b ound (6) captures the correct asymptotics of λ + c : Conjecture 1.2. λ + c ( d ) ∼ e 2 d as d → ∞ . 1.3 The hard-core mo del on discrete tori and other regu- lar bipartite graphs On a ﬁnite simple regular bipartite graph G = ( V , E ) , with bipartition ( E , O ) , w e seek to show that samples from the hard-core model at high fugacity tend to concentrate in one of the bipartition classes. T o this end, for each σ ∈ Ω G hc , consider the o ccupation coun ts of the tw o bipartition classes | σ E | , | σ O | and the total o ccupation count | σ | . W e also note the trivial low er b ound, obtained by requiring that | σ E | = 0 (or that | σ O | = 0 ), Z G ≥ (1 + λ ) 1 2 | V | . (8) 1.3.1 Discrete tori W e now detail our long-range order results for the discrete tori graphs Z d L , deﬁned as follows: Let Z L denote b oth the set of residue classes Z /L Z and the cycle graph of length L with vertex set Z /L Z . The Cartesian p ow er Z d L refers b oth to ( Z /L Z ) d and the discrete torus graph on this vertex set (with edge set {{ v , v + e i } : v ∈ ( Z /L Z ) d , i ∈ { 1 , . . . , d }} where the e i denote standard basis v ectors. T o make sure that the graph is simple, we identify parallel edges; as a result, Z d L is 2 d -regular when L ≥ 3 and d -regular when L = 2 ). The hard-core model on the hyp er cub e gr aph Z d 2 has attracted m uch attention in the literature. A seminal work b y Korsh uno v–Sap ozhenk o [KS83] (see [Gal19] for an exp ository note in English) sho wed that | Ω Z d 2 hc | ∼ 2 √ e 2 1 2 | Z d 2 | , (9) corresp onding to the exp ectation that for λ = 1 , typical indep enden t sets are con tained in either the even or o dd partite classes, with rare defects. A further breakthrough was achiev ed by Kahn [Kah01] who show ed, in a suitable sense, that this kind of long-range order arises already for λ = o (1) as d → ∞ . This w ork was the ﬁrst to introduce the p o w erful entrop y method to this problem, follo wing Kahn–La wrenz [KL99]. This w as signiﬁcan tly strengthened by Galvin [Gal11], who prov ed long-range order when λ > C log d d 1 / 3 . The state-of-the-art was 4 obtained recently b y Jenssen–Malekshahian–P ark [JMP26], who prov ed long- range order for λ > C log 2 d d 1 / 2 . In the other direction, the b ound (4) applies to sho w the absence of long-range order at fugacities λ ≤ λ d ∼ e d as d → ∞ . Our work establishes long-range order for the hypercub e graph at fugacities of order λ > C log d d , th us determining the critical fugacity up to the log factor. The result applies, more generally , to the discrete tori graphs Z d L (yielding long- range order for L ﬁxed as d → ∞ ), and this will play a role (with L = 6 ) in our pro of of Theorem 1.1. W e state the result as a corollary , as it is derived (in Section 11) from our main result for ﬁnite graphs, Theorem 1.7 b elo w. Corollary 1.3. F or e ach C 0 > 0 ther e exist C 1 , C 2 > 0 such that for e ach dimension d ≥ 2 and even L ≥ 2 : The event B : =  min {| σ E | , | σ O |} > L d +1 d C 0  ∪      | σ | − L d 2 λ 1 + λ     > L d +1 d C 0  (10) satisﬁes µ Z d L ( B ) ≤ ζ Z d L ( B ) (1 + λ ) L d / 2 ≤ (1 + λ ) − L d d C 2 (11) whenever λ > C 1 log d d . The corollary sho ws that indep enden t sets sampled from the hard-core mo del are ordered, in the sense of b eing mostly contained in a single partite class, asymptotically almost surely as d → ∞ for a ﬁxed L . Theorem 1.7 also yields the following b ound, complementing the trivial (8): Corollary 1.4. Ther e exist C, c > 0 such that for every λ > 0 , dimension d ≥ 1 and even L ≥ 2 , 1 L d log Z Z d L ≤ 1 2 log(1 + λ ) + C d (1 + λ ) − cd + C L d . (12) W e remark that m uch more precise b ounds than (12) w ere sho wn for the hyper- cub e in some of the ab o ve-cited w orks in restricted fugacity regimes, starting with (9). The widest fugacit y regime for this type of b ounds is achiev ed in [JMP26, Theorem 5.3] (con tinuing up on [JP20]), applying when λ > C log 2 d d 1 / 2 (and λ = O (1) ). F or Z d L , L > 2 even, see [JK23]. 1.3.2 Regular bipartite graphs Our main result on ﬁnite graphs dep ends on tw o expansion prop erties, on the lo cal and global scales. On the global scale, the expansion of G = ( V , E ) is measured by the Che e ger c onstant : h ( G ) : = min  | ∂ A | | A | : A ⊂ V , 0 < | A | ≤ | V | 2  , (13) 5 where ∂ A denotes the edge boundary of A in G . On the local scale, w e measure expansion in terms of the follo wing deﬁnition (see Section 2 for graph notations): Deﬁnition 1.5. Let G = ( V , E ) b e a ﬁnite δ -regular graph. Sa y that G sat- isﬁes the lo c al exp ansion pr op erty with p ar ameters C LE , M LE > 0 if there is a probabilit y distribution ov er c onne cte d subgraphs T of G with at least one edge suc h that when T is sampled from this distribution: 1. | E | · P ( uv ∈ E ( T )) ≤ M LE for each uv ∈ E , 2. | V | · P ( N ( v ) ∩ V ( T )  = ∅ ) ≥ δ M LE /C LE for each v ∈ V . While this deﬁnition is quite ad ho c, it is conv enien t to verify for the torus graphs Z d L , and is suited for our needs. One may chec k that necessarily C LE ≥ 1 2 and that there is no loss in generality in requiring T to be a tree. Any δ -regular ﬁnite graph G satisﬁes the lo cal expansion prop ert y with parameters C LE = M LE = 1 b y letting T be the subgraph of G consisting of a single uniformly chosen edge, though we will often seek to satisfy the prop erty with a larger v alue for M LE , while keeping C LE b ounded. The following lemma, which is not used in the pap er, sheds additional light on the local expansion prop ert y by showing that, at least on a bipartite G , lo cal expansion follows from “quantitativ e transience” of the simple random w alk. Lemma 1.6. L et G = ( V , E ) b e a ﬁnite δ -r e gular bip artite gr aph. L et C 0 ≥ 1 and let M 0 ≥ 2 inte ger. Supp ose that for every v ∈ V , a simple r andom walk W 0 , . . . , W M 0 − 1 starte d at W 0 = v , satisﬁes E |{ i ≥ 1 : W i = v }| ≤ ( C 0 − 1) /δ . Then G satisﬁes the lo c al exp ansion pr op erty with p ar ameters C LE = C 0 and M LE = M 0 , by taking T to b e the tr ac e of the walk W . W e ma y now state our main result for ﬁnite graphs. The ﬁrst part of the theorem giv es an upp er b ound on the partition function, complementing the trivial lo wer b ound (8), while the second part indicates long range order at suﬃcien tly high fugacities. Theorem 1.7. Ther e exist C , c > 0 such that the fol lowing holds for every λ, C LE , M LE > 0 and inte ger δ ≥ 2 . L et G = ( V , E ) denote a ﬁnite simple δ -r e gular bip artite gr aph, with bip artition ( E , O ) . Supp ose that G satisﬁes the lo c al exp ansion pr op erty with p ar ameters C LE , M LE . Then, ﬁrst, 1 | V | log Z G ≤ 1 2 log(1 + λ ) + C C LE δ  (1 + λ ) − δ C C LE + 1 M LE  . (14) Se c ond, for r > 0 , if log(1 + λ ) > C C LE δ max  log  C δ h ( G ) · | V | r  , δ h ( G ) · | V | r · 1 M LE  (15) then µ G (min {| σ E | , | σ O |} > r ) ≤ ζ G (min {| σ E | , | σ O |} > r ) (1 + λ ) | V | / 2 ≤ (1 + λ ) − c h ( G ) δ r . (16) 6 R emark 1.8 . The in termediate term in (16) may b e harnessed to study the distribution of indep enden t sets of a ﬁxed size N < | V | 2 . This may b e done as follo ws: Let µ G N b e the uniform distribution ov er Ω G hc ,N = { σ ∈ Ω G hc : | σ | = N } . Then, for every even t B , by noting that | Ω G hc ,N | ≥  | V | / 2 N  (similarly to (8)) and c ho osing λ to satisfy | V | 2 · λ 1+ λ = N , µ G N ( B ) ≤ | B ∩ Ω G hc ,N |  | V | / 2 N  = ζ G ( B ∩ Ω G hc ,N ) λ N  | V | / 2 N  = ζ G ( B ∩ Ω G hc ,N ) (1 + λ ) | V | / 2 P ( Bin ( | V | 2 , λ 1+ λ ) = N ) ≤ C p | V | ζ G ( B ) (1 + λ ) | V | / 2 , (17) where Bin ( n, p ) represents a binomial random v ariable with n trials and success probabilit y p , and we hav e con trolled the probability that it equals its exp ec- tation. This allows to bound µ G N (min {| σ E | , | σ O |} > r ) via Theorem 1.7 (or Corollary 1.3), for suitable ranges of N and r . R emark 1.9 (Sharpness) . In Subsection 14.2 we show that the second part of Theorem 1.7 cannot b e improv ed without further assumptions. Precisely , we pro vide examples of graphs (with expansion parameters similar to those of the h yp ercub e) for which Theorem 1.7 captures the correct order of magnitude of the threshold fugacity for long-range order (in a natural sense deﬁned there). The following lemma describes the prop erties of discrete tori used (in Subsection 11) to derive Corollary 1.3 from Theorem 1.7. Lemma 1.10. The torus Z d L , for al l inte gers d ≥ 1 , L ≥ 2 , satisﬁes the lo c al exp ansion pr op erty with C LE = 12 and M LE = 6 L d d , and has Che e ger c onstant satisfying h ( Z d L ) ≥ 1 L . 1.4 Outline Section 2 establishes basic notation. Section 3 discusses entrop y and the free energy functional which are at the heart of our arguments for ﬁnite graphs. In Part I w e prov e our main result for ﬁnite graphs, Theorem 1.7: First, global expansion is used in Section 4 to reduce to the k ey Prop osition 4.2, which b ounds the free energy functional of a probability measure on indep enden t sets in terms of a kind of exp ected “interface energy” E Φ of a sampled conﬁgura- tion. An imp ortan t idea in the proof of the proposition, sparse exp osure, is in tro duced in Section 5, reducing the pro of to an estimate of “gain” (Section 6) and “loss” (Section 7) terms in terms of E Φ . The loss term is controlled via the lo cal expansion prop ert y , which is used to pro duce an eﬃcient enco ding of the information revealed by the sparse exp osure. T he pro of of the prop osition is completed in Section 8, where it is also commented that the pro of simpliﬁes for sp ecial graphs, including the discrete tori Z d L . P art I I includes the proofs inv olv ed in specializing our results to the case of Z d L . This includes the proof of the lo cal expansion prop ert y (Section 9) and Cheeger constan t b ound (Section 10) for the discrete tori graphs Z d L , as stated in Lemma 7 1.10. It also includes Section 11, where Corollary 1.3 on long-range order on Z d L is deduced from Theorem 1.7 (relying also on deviation bounds for the binomial distribution). P art I II studies the hard-core mo del on the lattice Z d , proving Theorem 1.1. W e start in Section 12 b y recalling the chessboard estimate and its basic prop erties and establishing a comparison inequality among chessboard seminorms on tori of diﬀerent side lengths (Lemma 12.3). These are applied in Section 13 to prov e Theorem 1.1 via a Chessboard-Peierls argumen t driven by our long-range order result (Theorem 1.7) for the discrete torus graph Z d L with the ﬁxed side length L = 6 (Corollary 1.3 also suﬃces for proving Theorem 1.1, but we use Theorem 1.7 instead, as it gives a more precise quantitativ e b ound in (59)). W e exp ect that our argument for transferring results from tori of ﬁxed side length to the lattice can b e adapted to other reﬂection p ositive spin systems. The pap er concludes in P art IV with further discussion, including connections with earlier works and op en questions. App endix A presents the pro of of Lemma 1.6 while App endix B provides tw o pro ofs of the generalized Shearer’s inequalit y , Prop osition 3.1. 2 Notation Constan ts: The symbols c, C stand for suﬃcien tly small/large universal p osi- tiv e constan ts, and may tak e diﬀeren t v alues in diﬀerent expressions, even within the same line (i.e. they serve as an alternative to the p opular O and Ω nota- tions). This applies when c, C app ear without modiﬁers—symbols suc h as c α , C ′ are not sub ject to this conv ention. Natural num b ers: Denote N := { 1 , 2 , 3 , . . . } . Sums and restrictions: F or X ∈ [0 , ∞ ) A w e denote | X | : = P v ∈ A X v . When B ⊂ A , write X B for the restriction ( X v ) v ∈ B . W e abuse notation by conﬂating X v and X { v } . Probabilit y: W e use P and E to denote probabilities and exp ectations resp ec- tiv ely . The same v ariable will app ear in a light fon t when it is deterministic and in a b old font when it is random. The v ariable σ , standing for a conﬁguration, is sp ecial in that it is sometimes suppressed in the notation, so that we write f instead of f ( σ ) and f instead of f ( σ ) . F or an even t E , we denote by 1 E its indicator random v ariable. All of our probability spaces will b e ﬁnite, except in the discussion of inﬁnite- v olume Gibbs measures. Graphs: F or a graph H , we denote its vertex set and edge set by V ( H ) , E ( H ) resp ectiv ely . Edges are undirected. F or brevity , w e denote an edge { u, v } by uv . W e require all graphs in this pap er to be simple, i.e., to ha ve no m ultiple edges or self loops (this is essential as in the context of the hard-core mo del the regularit y 8 of the graph would b e meaningless if parallel edges are allow ed). In addition, when disc ussing δ -regular graphs, we alwa ys assume that δ ≥ 2 . The edge b oundary of a subset A ⊂ V ( H ) is denoted ∂ A = { uv ∈ E ( H ) : u ∈ A, v / ∈ A } . The neighborho od of a vertex v ∈ V ( H ) is denoted N ( v ) = { u ∈ V ( H ) : uv ∈ E ( H ) } . The symbol G alwa ys denotes a graph with V = V ( G ) and E = E ( G ) . When- ev er a bipartite graph is considered, its chosen bipartition classes are denoted E , O . T ori: The notation Z L and Z d L w ere introduced in Subsection 1.3. Single-site free energy : In the context of the hard-core mo del with fugacit y λ , we denote ˜ λ : = log(1 + λ ) . (18) 3 En trop y and the free energy functional 3.1 Shannon en trop y Let X b e a random v ariable. W e denote the Shannon entrop y of X b y S ( X ) = X x P ( X = x ) log 1 P ( X = x ) . (19) In our use of conditional entrop y , we distinguish conditioning on a random v ariable from conditioning on an even t. Precisely , given an additional random Y and an even t E (of non-zero probability—this will b e tacitly assumed b elo w when conditioning), we deﬁne the (exp ected) entrop y of X conditioned on Y and E as S ( X | Y ; E ) = X x,y P ( Y = y , X = x | E ) log 1 P ( X = x | Y = y , E ) . (20) W e may drop either Y or E by defaulting Y to b e a constant R V, and defaulting E to b e the whole probabilit y space. Note that typically S ( X | E )  = S ( X | 1 E ) . The deﬁni tion implies that, for random X , Y , Z and even t E , S ( X | Y , Z ; E ) = X y P ( Y = y | E ) · S ( X | Z ; E , Y = y ) . (21) W e shall use the basic prop erties of Shannon entro py as in tro duced in e.g. [CT05, AS08]. W e use the following generalized form of the entrop y chain rule: S ( X , Y | Z ; E ) = S ( X | Y , Z ; E ) + S ( Y | Z ; E ) . (22) 9 W e also use the fact that conditional entrop y is larger when c onditioning on less information: S ( X | Y ; E ) ≤ S ( X | f ( Y ); E ) (23) W e abbreviate S ( p ) for S ( X ) when p = P ( X = 1) = 1 − P ( X = 0) . W e note the following conv enient inequality: S ( x ) ≤ x log e x . (24) A standard fact is that the entrop y is subadditive S ( X , Y ) ≤ S ( X ) + S ( Y ) . This is signiﬁcantly extended by the following. Prop osition 3.1 (generalized Shearer inequality) . L et J b e a ﬁnite index set. L et ( X j ) j ∈ J b e r andom variables. Supp ose that K is a r andom subset of J , indep endent of X , with P ( j ∈ K ) ≥ p for e ach j ∈ J . Then S ( X ) ≤ 1 p S (( X j ) j ∈ K | K ) . (25) This is a straigh tforward generalization and rephrasing of the standard Shearer’s inequalit y [CGFS86], see App endix B. 3.2 The free energy functional Let Ω b e a ﬁnite set. Let H : Ω → ( −∞ , ∞ ] , not identically ∞ (termed the Hamiltonian). Deﬁne a measure ζ H on Ω by ζ H ( σ ) : = e −H ( σ ) . Denote Z H : = ζ H (Ω) and the (Gibbs) probabilit y measure µ H : = ζ H Z H . Let σ b e a random v ariable sampled from an arbitr ary probability measure on Ω . Deﬁne the (negative) free energy functional of σ as I H ( σ ) := S ( σ ) − E H ( σ ) . (26) In analogy with conditional entrop y , we deﬁne conditional versions of the free energy functional by I H ( σ | X ; E ) : = S ( σ | X ; E ) − E [ H ( σ ) | E ] . (27) The following fact, whic h ma y be v eriﬁed directly , implies that I H is maximized when σ is sampled from µ H , see e.g. [Geo11, Chapter 15] or [FV17, Lemma 6.74]. F act 3.2 (V ariational principle) . L et σ b e sample d fr om an arbitrary pr ob ability me asur e µ on Ω . Then I H ( σ ) = log Z H − d KL ( µ ∥ µ H ) . (28) wher e d KL ( µ ∥ µ H ) = P σ ∈ Ω µ ( σ ) log µ ( σ ) µ H ( σ ) ≥ 0 stands for the Kul lb ack–L eibler diver genc e. 10 In the sequel, we use the following cons equence: when σ is sampled from µ H , for each even t E ⊂ Ω , log ζ H ( E ) = I H ( σ | E ) . (29) This follows from F act 3.2 by noting that I H ( σ | E ) = I H| E ( σ | E ) = log Z H| E = log ζ H ( E ) , as µ H conditioned on E is the Gibbs measure of the restriction of the Hamiltonian H to E . 3.3 The free energy functional of the hard-core mo del W e now specialize the free energy functional to the hard-core mo del on a graph G . Let λ > 0 . W e set Ω = Ω G hc and H ( σ ) = −| σ | log λ , so that the measure µ H is the hard-core measure µ G deﬁned in (3). F or brevity , in the rest of the pap er, w e will simply write I for this I H , omitting mention of the graph G and the fugacity λ . W e thus hav e that for any random σ ∈ Ω G hc , I ( σ ) = S ( σ ) + (log λ ) E | σ | . (30) Giv en a subset A ⊂ V , the restriction σ A is a random indep enden t set in the induced subgraph G [ A ] (i.e., σ A ∈ Ω G [ A ] hc ), giving meaning to I ( σ A ) . Moreov er, since the term E | σ | in (30) is additive in the domain of σ , it follows that I inherits the subadditivity of S , i.e., I ( σ A ∪ B ) ≤ I ( σ A ) + I ( σ B ) for disjoint A, B ⊂ V . (31) In addition, the generalized Shearer inequalit y of Prop osition 3.1 applies with I replacing S , with ( σ v ) v ∈ V taking the place of ( X j ) j ∈ J and with the cav eat that the assumption P ( j ∈ K ) ≥ p of that prop osition is replaced by the e quality P ( v ∈ K ) = p for all v ∈ V (as log λ may b e negativ e). F or v ∈ V , we hav e the basic bound I ( σ v ) ≤ ˜ λ whic h may b e view ed as a special case of the v ariational principle, F act 3.2, with ˜ λ = log(1 + λ ) as in (18). The follo wing is a stronger, conditional version. Prop osition 3.3. If σ v ≤ X for a r andom X ∈ { 0 , 1 } then, for r andom Y and event E , I ( σ v | X , Y ; E ) ≤ ˜ λ E [ X | E ] . (32) Pr o of. T o see this, note ﬁrst that I ( σ v | X , Y ; E ) = P ( X = 0 | E ) I ( σ v | Y ; E , X = 0)+ P ( X = 1 | E ) I ( σ v | Y ; E , X = 1) by (21) and then use that I ( σ v | Y ; E , X = 0) = 0 by (30) and the fact that σ v = 0 when X = 0 and that I ( σ v | Y ; E , X = 1) ≤ ˜ λ . 11 P art I Long-range order for regular bipartite ﬁnite graphs 4 Coarse graining and global expansion In this section we deduce Theorem 1.7 from Proposition 4.2 b elo w, whose pro of is the sub ject of the next sections. Let G b e a δ -regular bipartite ﬁnite graph. Recall our con ven tion that G = ( V , E ) with bipartition classes E , O ⊂ V . When applicable, w e will use the con ven tion that v denotes and elemen t of E while u denotes an element of O . Giv en a conﬁguration σ ∈ Ω G hc w e deﬁne a function φ whic h we like to think of as a “coarse grained” or “smo othed” v ersion of σ . On the even side, deﬁne: φ v : = Y u ∈ N ( v ) (1 − σ u ) ∀ v ∈ E , (33) so that φ v is the indicator function of the even t that all neigh b ors of v are v acant in σ (so that v may b e o ccupied in σ ). W e also deﬁne tw o useful extensions of φ to the o dd side φ u : = Ma jority( φ v : v ∈ N ( u )) ∀ u ∈ O , (34) ⌢ φ u : = 1 δ X v ∈ N ( u ) φ v ∀ u ∈ O , (35) breaking ties arbitrarily . Also, still given σ ∈ Ω G hc , deﬁne Φ : = 1 | E | X uv ∈ E | φ u − φ v | = 2 | V | X u ∈O min { ⌢ φ u , 1 − ⌢ φ u } (36) as a measure of “roughness” for φ (and by pro xy , also for σ ) or a measure of “the (normalized) total (h yp er-)surface area of domain w alls b et ween the even- o ccupied phase and the o dd-occupied phase”. 4.1 Using the global expansion Within Section 4, denote M : = min {| σ E | , | σ O |} , (37) and recall the Cheeger constan t h ( G ) from (13). T o prov e Theorem 1.7, w e need to b ound the even t {M > r } . The following lemma will allow us to work in terms of Φ instead of M . 12 Lemma 4.1. L et G b e a ﬁnite δ -r e gular bip artite gr aph. L et σ ∈ Ω G hc . Then M ≤ δ 2 h ( G ) Φ | V | . (38) Pr o of. Introduce M ′ : = min { X v ∈E φ v , X v ∈E (1 − φ v ) } . (39) Note that | σ E | = P v ∈E σ v ≤ P v ∈E φ v since by the hard-core restriction, for eac h v ∈ E , σ v = 1 implies φ v = 1 . Also, | σ O | = P u ∈O σ u ≤ P v ∈E (1 − φ v ) by a double counting argument: δ X u ∈O σ u = X u ∈O X v ∈ N ( u ) σ u ≤ X u ∈O X v ∈ N ( u ) (1 − φ v ) = δ X v ∈E (1 − φ v ) (40) since for adjacent u ∈ O , v ∈ E , if σ u = 1 then φ v = 0 . Th us M ≤ M ′ . No w let A b e the smaller of the tw o sets { v ∈ V : φ v = 0 } and { v ∈ V : φ v = 1 } . Note | A | ≥ M ′ . By the deﬁnition of the Cheeger constant (13), h ( G ) ≤ | ∂ A | | A | = | E | Φ | A | ≤ | E | Φ M ′ . Th us M ≤ M ′ ≤ δ 2 h ( G ) Φ | V | . 4.2 Bounding the free energy functional by E Φ and de- duction of the main result Under suﬃciently strong lo cal expansion as quantiﬁed b y Deﬁnition 1.5, we show that if σ is distributed such that E Φ is suﬃcien tly large, then E Φ con tributes a “surface tension” term to the free energy functional I ( σ ) . Recall from (18) that ˜ λ = log(1 + λ ) . Prop osition 4.2. Ther e exist C, c > 0 such that the fol lowing holds. L et G b e a ﬁnite δ -r e gular bip artite gr aph, satisfying the lo c al exp ansion pr op erty with p ar ameters C LE , M LE . L et σ b e sample d fr om an y measure on Ω G hc . L et λ > 0 . Then 1 | V | I ( σ ) − 1 2 ˜ λ ≤ − 1 4 ˜ λ E Φ + C C LE δ  E Φ log e E Φ + 1 M LE  . (41) Conse quently, I ( σ ) − ˜ λ | V | 2 ≤ − c ˜ λ | V | E Φ (42) whenever ˜ λ > 9 C C LE δ max  log e E Φ , 1 M LE E Φ  . (43) This prop osition suﬃces for the Pr o of of The or em 1.7. The free energy b ound (14) is obtained as follows: Let σ b e sampled from µ G . Note that log Z G = I ( σ ) by the v ariational principle, 13 F act 3.2. The b ound follo ws from Proposition 4.2 b y maximizing the righ t-hand side of (41) ov er E Φ ∈ [0 , 1] . The ﬁrst inequality of (16) follo ws from (8). F or the second inequality , let σ b e sampled from µ G conditioned on the even t M > r . By (29), it holds that log ζ G ( M > r ) = I ( σ ) . Since E M > r , Lemma 4.1 implies that | V | h ( G ) r > 2 δ E Φ . T ogether with the theorem’s assumption (15), for suﬃciently large C , this implies the assumption (43) of Prop osition 4.2. W e get the second inequalit y as follo ws: log ζ G ( M > r ) = I ( σ ) ≤ ˜ λ  | V | 2 − c | V | E Φ  ≤ ˜ λ  | V | 2 − c h ( G ) δ r  = log h (1 + λ ) | V | / 2 − c h ( G ) δ r i . The remainder of P art I is about proving Prop osition 4.2. The pro of ma y b e simpliﬁed for a certain family of graphs G , which includes the tori Z d L . The simpliﬁcations are describ ed in the ﬁnal Subsection 8.1, and it may b e b eneﬁcial to keep them in mind, esp ecially on a ﬁrst read. 5 Sparse exp osure Our strategy for upp er b ounding I ( σ ) in Proposition 4.2 starts with the next lemma which provides a useful decomp osition as a sum of three terms, which w e will b ound separately . This decomp osition holds in general, without assump- tions on expansion. The random set A app earing in it may b e thought of as a set on which we “reveal” part of the information, sp eciﬁcally φ A , providing a starting p oin t for our later b ounds. It may b e useful to keep in mind that in our in tended application the densit y of B in O will b e at least a universal constan t, and the density of A in E will b e of order 1 /δ . Lemma 5.1. L et G b e a ﬁnite δ -r e gular bip artite gr aph. L et σ b e sample d fr om an y measure on Ω G hc . L et A ⊂ E and B ⊂ O b e r andom subsets, such that ( A , B ) is indep endent of σ and P ( u ∈ B ) is the same for al l u ∈ O and non-zer o. Denote by s := P ( u ∈ B ) its c ommon value. Then I ( σ ) ≤ I ( σ E | σ O ) + 1 s I ( σ B | B , A , φ A ) + 1 s S ( φ A | A ) . (44) Pr o of. It suﬃces to justify the follo wing steps: I ( σ ) = I ( σ E | σ O ) + I ( σ O ) ≤ I ( σ E | σ O ) + 1 s I ( σ B | B ) = I ( σ E | σ O ) + 1 s [ I ( σ B | B , A , φ A ) + S ( φ A | A )] . (45) 14 First, expand the deﬁnition of I (30) in all of its o ccurrences. F or the (log λ ) E | σ | terms, w e hav e equalities throughout, using the assumption that s = P ( u ∈ B ) for all u . It remains to pro ve the displa y ab o ve with all occurrences of I replaced b y S . The inequality follows from the generalized Shearer’s inequality (using that s ≤ P ( u ∈ B ) for all u ). The t wo equalities follo w from the en tropy c hain rule (22), where for the second equality w e also use that S ( σ B | B ) = S ( σ B | B , A ) and S ( φ A | A ) = S ( φ A | B , A ) , b oth of which follow, using the deﬁnition of entrop y , from the fact that ( A , B ) is indep enden t of ( σ , φ ) . Among the three terms app earing in the right-hand side of (44), we regard the ﬁrst t wo as “gain terms” whose b ound will improv e when E Φ is large, and the third term as a “loss” or error term which w e will seek to control suﬃciently tigh tly . The next t wo sections are devoted to b ounding these terms. W e p oin t out the sp ecial case that B = O deterministically (and thus s = 1 ). In this case we get that (44) is an e quality . In addition, the intuition that w e are rev ealing φ A is more obvious, and there is no need to apply Shearer’s inequalit y . An application of this sp ecial case is describ ed in Subsection 8.1. 6 The gain terms The next lemma bounds the ﬁrst t wo terms on the right-hand side of (44) (the gain terms). First, it is shown in (46) that they are b ounded by a sum of lo cal expressions. Second, w e bound these lo cal terms b y relying on the follo wing prop ert y of the distribution of ( A , B ) . W e sa y that ( A , B ) satisﬁes the two uniform neighb ors pr op erty if the follo wing holds for every u ∈ O : The distribution of A , conditioned on u ∈ B , can be coupled with ( v u, 1 , v u, 2 ) , a pair of independently uniformly sampled neighbors of u , such that { v u, 1 , v u, 2 } ⊂ A . Lemma 6.1. Under the assumptions of L emma 5.1: I ( σ E | σ O ) + 1 s I ( σ B | B , A , φ A ) − ˜ λ | V | 2 (46) (1) ≤ ˜ λ X u ∈O E   Y v ∈ N ( u ) ∩ A (1 − φ v ) + ⌢ φ u − 1 | u ∈ B   (47) (2) ≤ − ˜ λ X u ∈O E [ ⌢ φ u (1 − ⌢ φ u )] (3) ≤ − 1 4 ˜ λ | V | E Φ , (48) wher e we assume that ( A , B ) satisﬁes the two uniform neighb ors pr op erty for ine quality (2). Pr o of. W e ﬁrst prov e inequality (1). F or the ﬁrst term on the LHS, by the subadditivit y (31) of I , Prop osition 3.3, and the δ -regularity of G , w e hav e I ( σ E | σ O ) ≤ X v ∈E I ( σ v | φ v ) ≤ X v ∈E ˜ λ E φ v = ˜ λ X u ∈O E ⌢ φ u . (49) 15 F or the second term, I ( σ B | B , A , φ A ) ≤ X u ∈O I ( σ { u }∩ B | B , A , φ A ) ( By (21) and I ( σ { u }∩ B | B , A , φ A ; u / ∈ B ) = 0) = X u ∈O P ( u ∈ B ) I ( σ u | B , A , φ A ; u ∈ B ) ( as s = P ( u ∈ B ) and by Prop osition 3.3 ) ≤ s X u ∈O ˜ λ P ( ∀ v ∈ N ( u ) ∩ A , φ v = 0 | u ∈ B ) = s ˜ λ X u ∈O E   Y v ∈ N ( u ) ∩ A (1 − φ v ) | u ∈ B   . Finally , for the third term, ˜ λ | V | 2 = ˜ λ P u ∈O 1 . W e turn to inequality (2), assuming the t wo uniform neigh b ors prop ert y . The inequalit y will b e sho wn for each u ∈ O separately . W e prov e the inequality for a ﬁxed φ , using only the randomness in ( A , B ) . This suﬃces since ( A , B ) is indep enden t of φ . Fix u ∈ O and φ . Let ( v u, 1 , v u, 2 ) b e as in the tw o uniform neighbors prop ert y , so that conditioned on u ∈ B they are a uniformly sampled pair of vertices in N ( u ) (not necessarily distinct). F or brevity , denote E u [ · ] := E [ · | u ∈ B ] . Then E u   Y v ∈ N ( u ) ∩ A (1 − φ v ) + ⌢ φ u − 1   ≤ E u  (1 − φ v u, 1 )(1 − φ v u, 2 ) + φ v u, 1 − 1  = E u  − φ v u, 2 (1 − φ v u, 1 )   by the tw o uniform neighbors property  = − ⌢ φ u (1 − ⌢ φ u ) . Finally , to see inequalit y (3). Observe that x (1 − x ) ≥ 1 2 min { x, 1 − x } for x ∈ [0 , 1] . Therefore by (36), X u ∈O ⌢ φ u (1 − ⌢ φ u ) ≥ 1 2 X u ∈O min { ⌢ φ u , 1 − ⌢ φ u } = 1 4 | V | Φ . 7 The loss term This section is devoted to the pro of of the follo wing Lemma 7.1, and is the only place where the lo cal expansion prop ert y (Deﬁnition 1.5) is used directly . Let G = ( V , E ) b e a ﬁnite δ -regular graph (not necessarily bipartite) satisfying the lo cal expansion prop ert y with parameters C LE , M LE . Let φ : V → { 0 , 1 } b e a random assignment of bits to the vertices of G (sampled from an arbitrary distribution). Note that w e do not assume that φ is the one in tro duced in Section 4, but this notation is suggestive, as the application of Lemma 7.1 16 b elo w is to the φ of Section 4. W e let Φ b e given by (36). The following lemma sa ys that given lo cal expansion, a random restriction of φ may b e “enco ded eﬃcien tly” if φ is suﬃciently “smo oth”. Lemma 7.1. L et A b e a r andom subset of V , indep endent of φ and satisfying P ( v ∈ A ) ≤ p for every v ∈ V . Then S ( φ A | A ) ≤ C LE | V |  2( p + 1 δ ) · S ( E Φ ) + log 2 δ M LE  . (50) Pr o of of L emma 7.1. Let T b e the random subgraph given by the lo cal expan- sion prop ert y , sampled indep enden tly of φ and A . Denote N ( T ) : = S v ∈ V ( T ) N ( v ) . Let A ⊂ V . Apply the generalized Shearer’s inequalit y (Theorem 3.1 with ( X j ) j ∈ J = φ A and K = N ( T ) ∩ A ), in the probabilit y space conditioned on A = A . This gives S ( φ A | A = A ) ≤ S ( φ N ( T ) ∩ A | T ; A = A ) min v ∈ A P ( v ∈ N ( T )) . (51) T o b ound the (exp ectation ov er A of the) numerator we introduce the edge set E ( T , A ) : = E ( T ) ∪ { uv ∈ E : u ∈ A , v ∈ V ( T ) } . Observ e that b y item 1 of the lo cal expansion prop ert y P ( uv ∈ E ( T )) ≤ 2 M LE / ( δ | V | ) for each uv ∈ E , and therefore also we hav e P ( u ∈ V ( T )) ≤ 2 M LE / | V | for each u ∈ V . the indep endence of T and A , and the assumption that max v P ( v ∈ A ) ≤ p , P ( uv ∈ E ( T , A )) ≤ P ( uv ∈ E ( T )) + P ( u ∈ V ( T )) P ( v ∈ A ) + P ( u ∈ A ) P ( v ∈ V ( T )) ≤ 4 M LE | V | ( 1 δ + p ) . (52) Note that E ( T , A ) spans a connected graph, whose vertex set contains N ( T ) ∩ A . Since φ tak es v alues in { 0 , 1 } , it follo ws that φ N ( T ) ∩ A ma y be deduced from ( | φ u − φ v | ) uv ∈ E ( T , A ) and φ v 0 ( T ) where v 0 ( T ) denotes an arbitrary v ertex of T designated as its ro ot. Th us, using that fact that T is indep enden t of ( φ , A ) S ( φ N ( T ) ∩ A | T , A ) ≤ S ( φ v 0 ( T ) | T , A ) + X uv ∈ E S ( | φ u − φ v | · 1 uv ∈ E ( T , A ) | T , A ) = S ( φ v 0 ( T ) | T , A ) + X uv ∈ E P ( uv ∈ E ( T , A )) S ( | φ u − φ v | ) ( b y (52), binary entrop y ) ≤ log 2 + 4 M LE | V | ( 1 δ + p ) X uv ∈ E S ( E | φ u − φ v | ) (Jensen’s in equalit y , 2 | E | = | V | δ ) ≤ log 2 + 2 δ M LE ( 1 δ + p ) S ( E Φ ) . (53) 17 Finally , S ( φ A | A ) = X A ⊂ V P ( A = A ) S ( φ A | A = A ) ( b y (51), and A ⊂ V ) ≤ X A ⊂ V P ( A = A ) S ( φ N ( T ) ∩ A | T ; A = A ) min v ∈ A P ( v ∈ N ( T )) = S ( φ N ( T ) ∩ A | T , A ) min v ∈ V P ( v ∈ N ( T )) and the lemma follows by substituting (53) and item 2 of the lo cal expansion prop ert y (noting that v ∈ N ( T ) ⇐ ⇒ N ( v ) ∩ V ( T )  = ∅ ). 8 Pro of of Prop osition 4.2 Pr o of. Let G be a ﬁnite δ -regular bipartite graph, satisfying the lo cal expansion prop ert y with parameters C LE , M LE . Let σ b e sampled from any me asur e on Ω G hc . Let λ > 0 . Let A ⊂ E be random, sampled indep enden tly of σ , with P ( v ∈ A ) = 1 /δ indep enden tly for each v ∈ E . Deﬁne B ⊂ O to b e { u ∈ O : | N ( u ) ∩ A | ≥ 2 } . Then the pair ( A , B ) satisﬁes the assumptions of Lemma 5.1 with s = P ( u ∈ B ) = 1 − (1 − 1 /δ ) δ − δ 1 δ (1 − 1 /δ ) δ − 1 ≥ c for all u ∈ O (using that δ ≥ 2 ). Lemma 6.1 applies to ( A , B ) as well, since ( A , B ) satisﬁes the tw o uniform neighbors prop ert y . Notice that φ , Φ , A , p = 1 δ satisfy the conditions of Lemma 7.1, and thus (50) h olds. W e now combine the three-term decomp osition with the bounds on the gain terms and the loss term: I ( σ ) − ˜ λ | V | 2 (b y Lemma 5.1) ≤ I ( σ E | σ O ) + 1 s I ( σ B | B , A , φ A ) − ˜ λ | V | 2 + 1 s S ( φ A | A ) . (Lemma 6.1, and 1 /s ≤ C ) ≤ − 1 4 ˜ λ | V | E Φ + C · S ( φ A | A ) (b y Lemma 7.1 and (24)) ≤ − 1 4 ˜ λ | V | E Φ + C C LE | V |  4 δ E Φ log e E Φ + log 2 δ M LE  whic h veriﬁes (41) (using a larger v alue of C ). The second part of the prop osition is an immediate consequence. 8.1 Simpliﬁed pro of in a sp ecial case The pro of of Prop osition 4.2 (whic h is the main ingredien t in the pro of of Theo- rem 1.7) admits a substantial simpliﬁcation in a sp ecial case, as we no w describe. A dominating tree of a graph is a subgraph which is a tree and whose vertex set is dominating (see Section 9 b elo w for the deﬁnition). Let q ∈ (0 , 1] and assume 18 that the graph G has a random dominating tree T (ha ving at least one edge) with max uv ∈ E P ( uv ∈ E ( T )) ≤ q . (54) Using this T in the deﬁnition of the lo cal expansion prop erty , we see that the prop ert y is satisﬁed with M LE = q | E | and C LE = δ M LE | V | (in fact, in the context of Deﬁnition 1.5, the last equalit y is equiv alen t to V ( T ) b eing almost surely dominating). The pro of of Prop osition 4.2 under the ab o v e assumption (and with these resulting C LE , M LE parameters) may b e simpliﬁed as follo ws: 1. Restrict Lemma 5.1 and Lemma 6.1 to the case that B = O determinis- tically (and thus s = 1 ). This circumv en ts the use of Shearer’s inequalit y in Lemma 5.1 and the conditioning on u ∈ B in Lemma 6.1. 2. Replace Lemma 7.1 b y the statemen t S ( φ V ( T ) | T ) ≤ q | E | · S ( E Φ ) + log 2 , where T is any random tree satisfying (54), sampled indep enden tly of φ . The proof is a v ariation on (53) where we put T instead of T , A , and V ( T ) instead of N ( T ) ∩ A , and we use (54) instead of (52). T o complete the pro of of Prop osition 4.2 in the sp ecial case, w e mo dify the c hoice of A in Section 8, b y taking A = V ( T 1 ) ∪ V ( T 2 ) with T 1 , T 2 b eing t wo indep enden t copies of the random dominating tree T . This leads to B = O . W e then chec k that the restricted Lemma 5.1 and Lemma 6.1 apply to this alternativ e c hoice of ( A , B ) and b ound S ( φ A | A ) by noting that S ( φ A | A ) ≤ 2 S ( φ V ( T ) | T ) and using the restricted Lemma 7.1. As an imp ortan t example, we point out that Z d L has a random dominating tree as ab ov e with q ≤ C d 2 b y the construction of Section 9 b elo w. Thus, the ab o ve sp ecial case of Prop osition 4.2 suﬃces when pro ving Corollary 1.3, Corollary 1.4, and consequently Theorem 1.1 (but not Theorem 1.7 in its full generality). In particular, this results in a pro of of these 3 results whic h do es not use Shearer’s inequalit y . P art I I Long-range order on discrete tori In this part, we prov e Lemma 1.10 (Section 9 and Section 10) and Corollary 1.3 (Section 11). Corollary 1.4 is an immediate consequence of Theorem 1.7 by substituting the parameters of Lemma 1.10. 9 Lo cal expansion for Z d L Let G b e a graph. F or A ⊂ V denote by A ∗ the set of vertices that are in A or hav e a neigh b or in A . A dominating set is a subset D ⊂ V , with D ∗ = V . 19 Deﬁne the domination numb er as γ ( G ) := min {| D | : D ⊂ V , D ∗ = V } . When a subgraph T of G is a tree and V ( T ) is dominating in G , we call T a dominating tr e e of G . Lemma 9.1. L et d ≥ 1 , L ≥ 2 inte gers. Then γ ( Z d L ) < 2 L d /d . Pr o of. Set r := max { s ∈ N : 2 s − 1 ≤ d } and denote d ′ = 2 r − 1 . Let H ⊂ Z d ′ 2 b e the Hamming co de [Wik25]. Since H is a p erfect, single error-correcting co de, it is a dominating set and | H | = 2 d ′ / ( d ′ + 1) . In the case that L is ev en, w e ma y simply choose D : =  ( v 1 , . . . , v d ) ∈ Z d L : ( v 1 mo d 2 , . . . , v d ′ mo d 2) ∈ H  as a dominating set for Z d L , noting that | D | = L d / ( d ′ + 1) < 2 L d /d . When L is o dd, denote D b : =  ( v 1 , . . . , v d ) ∈ { 0 , . . . , L − 1 } d : ( v 1 mo d 2 , . . . , v d ′ mo d 2) ∈ H + b  where b ∈ Z d ′ 2 . Note that here v 1 , . . . , v d are tak en to b e integers rather than residue classes mo dulo L , so that they can b e considered mo dulo 2 . The set D b view ed as a subset of Z d L is dominating for all b ∈ Z d ′ 2 . T aking b ∈ Z d ′ 2 uniformly random, it holds that E [ | D b | ] = | Z d L | · | H | / | Z d ′ 2 | = L d / ( d ′ + 1) < 2 L d /d. Th us there is a choice of b such that | D b | < 2 L d /d . Lemma 9.2. L et G b e a c onne cte d gr aph. T hen G has a dominating tr e e T of or der at most 3 γ ( G ) . Pr o of. Let D b e a dominating set in G with | D | = γ ( G ) . Construct a graph G 0 with vertex set D where t wo v ertices u, v ∈ D are adjacen t in G 0 iﬀ the distance from u to v in G is at most 3 . W e claim that G 0 is connected. Indeed, assume for contradiction that A is the vertex set of a connected comp onen t, and that B : = D \ A is non-empty . Then since D is a dominating set, A ∗ ∪ B ∗ = G (where ∗ is taken with resp ect to the connectivity of G ). By the connectedness of G and the fact that A ∗ , B ∗ are nonempty it follows that there is an edge uv ∈ E ( G ) with u ∈ A ∗ , v ∈ B ∗ . Th us there is u ′ ∈ A with u b eing equal or adjacen t to u ′ and similarly v ′ ∈ B that is equal or adjacen t to v . Th us u ′ , v ′ are at graph distance at most 3 from each other in G , so that u ′ v ′ ∈ E ( G 0 ) con tradicting the assumption. W e construct T as follows. T ake a spanning tree of G 0 and for each edge uv of it, pick a path of length at most 3 in G with endp oin ts u and v . Denote b y T ′ the connected subgraph of G formed by the union of these paths. Note that | V ( T ′ ) | ≤ 3 γ ( G ) and D ⊂ V ( T ′ ) . T ake T to b e an arbitrary spanning tree of T ′ . 20 Lemma 9.3. L et G b e an e dge tr ansitive gr aph, having a dominating tr e e of or der M with at le ast one e dge. Then G satisﬁes the lo c al exp ansion pr op erty with any M LE ≥ M and C LE ≥ δ M LE | V | . Pr o of. Let T b e a dominating tree of order M . Deﬁne a random connected subgraph T ⊂ G by acting on T by a random automorphism of G . Since G is edge transitive, it follows that item 1 is satisﬁed as | E ( T ) | = M − 1 ≤ M LE . F or item 2 the probability is 1 , since V ( T ) is alwa ys dominating, and thus the condition holds with any C LE ≥ δ M LE | V | . Let d ≥ 1 , L ≥ 2 in tegers. Combining Lemma 9.1 and Lemma 9.2 ab ov e, Z d L has a dominating tree of size at most 6 L d /d . Since Z d L is edge transitive, Lemma 9.3 implies that Z d L satisﬁes the lo cal expansion prop erty with M LE = 6 L d /d and C LE = 2 d · 6 L d /d L d = 12 (we ha ve δ = 2 d for L ≥ 3 and δ = d when L = 2 ). This gives the lo cal expansion part of Lemma 1.10. 10 Global expansion for Z d L The following lemma gives the global expansion part of Lemma 1.10. Lemma 10.1. h ( Z d L ) ≥ 1 L for inte gers d ≥ 1 , L ≥ 2 . Pr o of. The one-dimensional case is trivial: h ( Z L ) ≥ 4 L when L ≥ 3 and h ( Z L ) = 1 when L = 2 . T o go up in dimension, we use the fact that for every graph G , it holds that h ( G ) / 2 ≤ h ( G d ) ≤ h ( G ) ; see [CT98, Theorem 1.2] or [Til00, Section 3, Prop osition 1]. R emark 10.2 . The P oincaré inequality giv es the w eaker bound h ( Z d L ) ≥ c L 2 . W e note that our pro of of Theorem 1.1 only relies on the case L = 6 , making use of the inequality h ( Z d 6 ) ≥ c , for whic h the weak er b ound is also suﬃcient. 11 Pro of of Corollary 1.3 W e will use the follo wing lemma regarding the tails of the binomial distribution. Lemma 11.1. F or every n ∈ N , m ≥ 0 , and 0 < p < 1 , P ( | Bin( n, p ) − np | ≥ m ) ≤ 2 ( p (1 − p )) m 2 /n . Pr o of. W e ha ve P (Bin( n, p ) − np ≥ m ) ≤ exp  − m 2 n g ( p )  ≤ ( p (1 − p )) m 2 /n . 21 The ﬁrst inequality is [Ho e63, (2.2)], with the function g ( p ) := ( 1 1 − 2 p log 1 − p p 0 < p < 1 2 1 2 p (1 − p ) 1 2 ≤ p < 1 deﬁned in [Hoe63, (2.4)]. The second inequality follo ws b y noting (with elemen- tary calculus) that g ( p ) > log 1 p (1 − p ) . By symmetry of our low er b ound for g ( p ) , the same b ound applies for P (Bin( n, p ) − np ≤ − m ) = P (Bin( n, 1 − p ) − n (1 − p ) ≥ m ) . Pr o of of Cor ol lary 1.3. The ﬁrst inequality is simply (8). F or the second in- equalit y we b egin b y b ounding µ Z d L ( B ) and then deduce the required b ound on ζ Z d L ( B ) . Let C 0 > 0 . W rite G E := n | σ O | ≤ L d +1 4 d C 0 +1 o , G O := n | σ E | ≤ L d +1 4 d C 0 +1 o and B 0 := n    | σ | − L d 2 λ 1+ λ    > L d +1 d C 0 o . Note that for B from (10) we ha ve B ⊂ ( G E ∪ G O ) c ∪ B 0 = ( G E ∪ G O ) c ∪ ( B 0 ∩ G E ) ∪ ( B 0 ∩ G O ) . (55) W e now b ound the probabilit y of ( G E ∪ G O ) c . Set r = L d +1 4 d C 0 +1 and note that (15) follo ws from the assumption λ > C 1 log d d (for suﬃcien tly large C 1 ) b y Lemma 1.10. Th us (16) holds, giving µ Z d L (( G E ∪ G O ) c ) ≤ (1 + λ ) − c L d d C 0 +2 . W e proceed to b ound the probability of B 0 ∩ G E . Conditioned on σ O , the v ariable | σ E | is distributed as Bin( | φ E | , λ 1+ λ ) with φ deﬁned in (33). Note that L d 2 − 2 d | σ O | ≤ | φ E | ≤ L d 2 (since Z d L has degree b ound 2 d ). No w, since B 0 ∩ G E =      | σ | − L d 2 λ 1 + λ     > L d +1 d C 0  ∩ G E =      | σ E | − | φ E | λ 1 + λ + | σ O | − ( L d 2 − | φ E | ) λ 1 + λ     > L d +1 d C 0  ∩ G E ⊂      | σ E | − | φ E | λ 1 + λ     > L d +1 d C 0 −     | σ O | − ( L d 2 − | φ E | ) λ 1 + λ      ∩ G E ( | φ E | ≤ L d 2 − 2 d | σ O | , λ 1 + λ ≤ 1 ) ⊂      | σ E | − | φ E | λ 1 + λ     > L d +1 d C 0 − (2 d − 1) L d +1 4 d C 0 +1  ⊂      | σ E | − | φ E | λ 1 + λ     > L d +1 2 d C 0  w e conclude, using Lemma 11.1 with n = | φ E | , p = λ 1+ λ and m = L d +1 2 d C 0 , that µ Z d L ( B 0 ∩ G E ) ≤ 2 max n ≤ L d 2 ( p (1 − p )) m 2 /n ≤ 2 max n ≤ L d 2 (1 − p ) m 2 /n = 2 (1 + λ ) − 2 L d  L d +1 2 d C 0  2 = 2 (1 + λ ) − L d +2 2 d 2 C 0 . 22 Analogously , we get the same b ound for the probabilit y of B 0 ∩ G O . Combining all the bounds obtained so far, w e get µ Z d L ( B ) ≤ 5(1 + λ ) − c L d d 2 C 0 +2 . By Corollary 1.4 together with λ > C 1 log d d , we hav e Z Z d L (1 + λ ) L d / 2 ≤ C exp  L d C d (1 + C 1 log d d ) − cd  = C (1 + λ ) C L d 1 ˜ λd (1+ C 1 log d d ) − cd ≤ C (1 + λ ) C L d (1+ C 1 log d d ) − cd . Th us, ζ Z d L ( B ) (1 + λ ) L d / 2 = µ Z d L ( B ) Z Z d L (1 + λ ) L d / 2 ≤ 5 C (1+ λ ) − c L d d 2 C 0 +2 + C L d (1+ C 1 log d d ) − cd ≤ 5 C (1+ λ ) − c L d d 2 C 0 +2 ≤ (1+ λ ) − L d d C 2 where the second-to-last inequality uses that C 1 is large enough as a function of C 0 and the last inequality follows by taking C 1 , C 2 large enough as a function of C 0 and using that λ > C 1 log d d and L, d ≥ 2 . P art I I I Long-range order on Z d Our proof of the existence of m ultiple Gibbs measures (Th eorem 1.1) is an instance of the chessboard Peierls argument (see [FILS78, FILS80] where this term is introduced, to the b est of our knowledge). W e deﬁne contours based on a grid of cub es of side length ℓ = 3 , and use the chessboard estimate to relate the probabilit y for the appearance of a contour to the probability of a ﬁxed ev ent in the hard-core mo del on Z d 6 , which we b ound via Theorem 1.7. 12 The c hessb oard estimate In this section, w e set up the notation and prop erties required for our use of the c hessb oard es timate. Our presentation adapts the notation of [HP25]. The c hessb oard estimate is a consequence of the reﬂection p ositivit y of the hard-core measure, with resp ect to reﬂections through co ordinate hyperplanes p assing thr ough vertic es . W e do not give a pro of of the chessboard estimate or of the reﬂection p ositivit y prop ert y and refer to [FILS78], [Shl86], [Bis09], [FV17, Chapter 10], [PS19, Section 2.7.1], [Had22] for p edagogical references. W e note, how ever, that they are a consequence of the fact that the interactions of the hard-core mo del are nearest-neigh b or symmetric pair interactions (or more generally , reﬂection-inv ariant hypercub e interactions) and the results of this section generalize immediately to other mo dels with such interactions. 23 12.1 Deﬁnitions T o state the chessboard estimate, we make the following deﬁnitions, for each ℓ ∈ N , L ∈ 2 ℓ N and λ > 0 : • Recall that Z m = Z /m Z and that Z d m stands for b oth a set of vectors of residue class es and a torus graph, as deﬁned in Subsection 1.3. • Let D L/ℓ denote the dihedral group of L/ℓ elements. W e consider its action on R /L Z where rotations, indexed by n ∈ Z L/ 2 ℓ , act as τ v = v + 2 nℓ and reﬂections, also indexed by n ∈ Z L/ 2 ℓ , act as τ v = 2 nℓ − v . – The direct pro duct of d copies, D d L/ℓ , acts on the Cartesian pro duct ( R /L Z ) d in the natural wa y . Observe that [0 , ℓ ] d is a fundamental domain for this action. – Deﬁne the action of τ ∈ D d L/ℓ on { 0 , 1 } Z d L and its action on R { 0 , 1 } Z d L b y ( σ τ ) v : = σ τ v and ( τ f )( σ ) : = f ( σ τ ) . (56) • A function f : { 0 , 1 } Z d → R is called [0 , ℓ ] d -lo cal if f ( σ ) dep ends only on the restriction of σ to [0 , ℓ ] d ∩ Z d . – W e also regard such f as f : { 0 , 1 } Z d L → R by regarding σ ∈ { 0 , 1 } Z d L as L Z d -p eriodic functions on Z d . • Deﬁne the ( ℓ, L ) - c hessb oard seminorm of a [0 , ℓ ] d -lo cal function f b y ∥ f ∥ ℓ | L : = | D d L/ℓ | v u u u u t ζ Z d L    Y τ ∈ D d L/ℓ τ f    . (57) – W e ﬁnd it con v enient to mak e this deﬁnition with the non-normalized measure ζ Z d L (see (2)) instead of the (more common) normalized mea- sure µ Z d L . This is esp ecially useful for Lemma 12.3 b elo w. – The in tegral in (57) is necessarily non-negative b y reﬂection p ositiv- it y , so that ∥ f ∥ ℓ | L is well deﬁned and satisﬁes ∥ f ∥ ℓ | L ≥ 0 . (58) – The name ‘seminorm’ is justiﬁed b y Prop osition 12.2 b elow. 12.2 Prop erties Prop osition 12.1 (Chessb oard estimate) . L et ℓ ∈ N , L ∈ 2 ℓ N and λ > 0 . F or e ach tuple ( f τ ) τ ∈ D L/ℓ of [0 , ℓ ] d -lo c al functions it holds that ζ Z d L    Y τ ∈ D d L/ℓ τ f τ    ≤ Y τ ∈ D d L/ℓ ∥ f τ ∥ ℓ | L . 24 The next fact captures several basic prop erties of the chessboard seminorm ∥·∥ ℓ | L (in particular, justifying the name seminorm); see [HP25, Prop osition 3.3], or [Bis09, Lemma 5.9], for a pro of. F act 12.2 (Positiv e homogeneit y , triangle inequalit y and monotonicity) . L et ℓ ∈ N , L ∈ 2 ℓ N and λ > 0 . The mapping f 7→ ∥ f ∥ ℓ | L , wher e f r anges over [0 , ℓ ] d -lo c al functions, satisﬁes the fol lowing pr op erties: 1. Homogeneit y: ∥ αf ∥ ℓ | L = | α | ∥ f ∥ ℓ | L for α ∈ R . 2. T riangle inequality: ∥ f 0 + f 1 ∥ ℓ | L ≤ ∥ f 0 ∥ ℓ | L + ∥ f 1 ∥ ℓ | L . 3. Monotonicit y: ∥ g ∥ ℓ | L ≥ ∥ f ∥ ℓ | L whenev er g ≥ f ≥ 0 . The next lemma allows to compare the chessboard seminorm b et ween tori of diﬀeren t side lengths, when these side lengths are divisible by one another. W e w ould like to highlight this lemma as it is useful in our approach and we hav e not seen it in the s tandard references on reﬂection p ositivit y . W e arrived at it from a similar statement in [Cha15, equation (6)], which refers to [CW98, see equation (8)]. W e again emphasize that its pro of relies only on the fact that the interactions of the hard-core model are nearest-neighbor symmetric pair in teractions (or more generally , reﬂection-inv arian t hypercub e interactions). Lemma 12.3 (Comparing chessboard seminorms on diﬀerent tori) . L et ℓ ∈ N , L ∈ 2 ℓ N and λ > 0 . F or e ach [0 , ℓ ] d -lo c al function f , ∥ f ∥ ℓ | L +2 ℓ ≤ ∥ f ∥ ℓ | L . Pr o of. W e use the fact that when A is a p ositiv e semi-deﬁnite matrix k p T r( A k ) ≥ k +1 p T r( A k +1 ) for each k ∈ N . Consider the sequence n i = L d − i ( L +2 ℓ ) i ℓ d v u u u u t ζ Z d − i L × Z i L +2 ℓ    Y τ ∈ D d − i L/ℓ × D i ( L +2 ℓ ) /ℓ τ f    and note that ∥ f ∥ ℓ | L = n 0 and ∥ f ∥ ℓ | L +2 ℓ = n d . Thus it suﬃces to show that n i is decreasing with i . T o simplify the presentation w e will only illustrate this by pro ving that n d ≤ n d − 1 . W e will deﬁne a suitable transfer matrix B suc h that ζ Z d L +2 ℓ    Y τ ∈ D d ( L +2 ℓ ) L/ℓ τ f    = T r  ( B B T ) L +2 ℓ 2 ℓ  ζ Z 1 L × Z d − 1 L +2 ℓ    Y τ ∈ D 1 L/ℓ × D d − 1 ( L +2 ℓ ) /ℓ τ f    = T r  ( B B T ) L 2 ℓ  . Putting A = B B T and k = L 2 ℓ giv es n d ≤ n d − 1 . The matrix B is indexed by Ω Z d − 1 L +2 ℓ hc (conﬁgurations on hyperplanes). F or σ 0 , σ 1 ∈ Ω Z d − 1 L +2 ℓ hc , deﬁne Ξ( σ 0 , σ 1 ) : = { σ ∈ Ω P ℓ × Z d − 1 L +2 ℓ hc : σ { 0 }× Z d − 1 L +2 ℓ = σ 0 , σ { ℓ }× Z d − 1 L +2 ℓ = σ 1 } 25 B ( σ 0 , σ 1 ) : = Z Ξ( σ 0 ,σ 1 ) λ − | σ 0 | + | σ 1 | 2 Y τ ∈{ Id }× D d − 1 ( L +2 ℓ ) /ℓ τ f ( σ ) dζ P ℓ × Z d − 1 L +2 ℓ ( σ ) where P ℓ is the path graph with vertex set { 0 , . . . , ℓ } connected in order. 13 Long-range order on Z d In this section we pro ve Theorem 1.1, sho wing the existence of at least tw o distinct Gibbs measures for the hard-core mo del on the lattice Z d at fugacit y λ > C log d d . The theorem follo ws from the next lemma, in which w e claim the existence of a “symmetry-breaking lo cal observ able” f . The precise deﬁnition of f is given in Subsection 13.2.2, but one should think of f = 1 ( f = − 1 ) as indicating that σ is in the “o dd-(ev en-)o ccupied phase” in the cub e [0 , ℓ ] d , while the complementary ev ent f = 0 indicates that [0 , ℓ ] d is part of a “con tour” separating the tw o phases. W orking on Z d L , this observ able is mapp ed to other cub es ([0 , ℓ ] d + ℓs ) s ∈ Z d L/ℓ via the mappings τ ∈ D d L/ℓ . The lemma quantiﬁes the rarit y of con tours at high fugacities. Its pro of is enabled by the chessboard estimate and the result of Theorem 1.7 for the discrete torus graph Z d 2 ℓ . In tro duce the following notation: F or each s ∈ Z d L/ℓ denote by τ s ∈ D d L/ℓ the unique element which maps the fundamental domain [0 , ℓ ] d to [0 , ℓ ] d + ℓs . Lemma 13.1. Ther e ar e C, c > 0 such that for e ach ℓ, d ∈ N and λ > 0 ther e exists a [0 , ℓ ] d -lo c al function f : { 0 , 1 } Z d → {− 1 , 0 , 1 } with the fol lowing pr op erties: L et L ∈ 2 ℓ N . Supp ose that σ is sample d fr om µ Z d L . 1. Deterministically , for each s, t nearest neighbors in Z d L/ℓ , τ s f · τ t f ≥ 0 . 2. E [ f ] = 0 when ℓ is o dd. 3. If d ≥ 2 , ℓ = 3 and λ > C log d d , then for each A ⊂ Z d L/ℓ , P ( τ s f = 0 ∀ s ∈ A ) ≤  5 d (1 + λ ) − c λ (1+ λ ) d ℓ d  | A | . (59) R emark 13.2 . Item 3 holds for any ℓ ≥ 1 , how ever the constan t c deteriorates as ℓ → ∞ . T o reduce clutter we restrict to ℓ = 3 as this is the only v alue that w e need for the pro of (any ﬁxed o dd ℓ > 1 would do). W e prov e the lemma in Subsection 13.2. Let us now explain how it implies Theorem 1.1. 13.1 Pro of of Theorem 1.1 Pr o of. Let the dimension d ≥ 2 . Fix ℓ = 3 and supp ose that λ > C 0 log d d for C 0 suﬃcien tly large for the following arguments. Let f b e the function from Lemma 13.1. 26 Step 1 : Note that by taking C 0 large enough, the condition on λ for Part 3 of Lemma 13.1 holds. By our choice of ℓ and and assumption on λ , we may c ho ose C 0 large enough so that the base of the exp onen t on RHS of (59) is at most c 0 d − 2 where c 0 → 0 as C 0 → ∞ . It follows that for each L ∈ 2 ℓ N and each s ∈ Z d L/ℓ , max { µ Z d L ( f = 0) , µ Z d L ( f  = τ s f ) } ≤ C c 0 d − 2 (60) F or the even t { f = 0 } this holds by (59). F or { f  = τ s f } , this is a consequence of the standard Peierls argumen t, as we brieﬂy detail. T o a void discussing non-con tractible cycles, it is con venien t to consider the cub e subgraph P d L/ℓ − 1 of Z d L/ℓ . Here, P r is the path graph with v ertex set { 0 , . . . , r } connected in order. W e sa y that a set of v ertices A in P d L/ℓ − 1 is a minimal separating set if it separates 0 and s in P d L/ℓ − 1 and is minimal with resp ect to inclusion with this prop ert y . Let A n b e the family of minimal separating sets A with | A | = n . The P eierls argument relies on the fact that |A n | ≤ ( C d 2 ) n . (61) This fact implies (60) as follows: By Part 1 of Lemma 13.1, if τ s f  = f then there exists A ∈ ∪ n A n suc h that τ s f = 0 for all s ∈ A . The union b ound thus implies that µ Z d L ( τ s f  = f ) ≤ P A ∈A n P ( τ s f = 0 ∀ s ∈ A ) and (60) follows from (61) and Part 3 of Lemma 13.1, taking C 0 suﬃcien tly large. The b ound (61) is a fairly standard fact ab out the cub e P d L/ℓ − 1 . W e brieﬂy indicate the ingredien ts that go in to it: First, by [Tim12, Lemma 2], ev ery A ∈ ∪ n A n is connected in the graph ( P d L/ℓ − 1 ) + , where ( P d L/ℓ − 1 ) + is the graph whic h contains P d L/ℓ − 1 and adds edges b et ween opp osing vertices of each cycle of length 4 in P d L/ℓ − 1 . The factor d 2 in (61) stems from the fact that the maximal degree in ( P d L/ℓ − 1 ) + is of order d 2 . Second, if the graph distance b et w een A ∈ ∪ n A n and { 0 , s } is large then A itself has to be large (here we use that d ≥ 2 ). F or instance, it is simple to see that if the distance is at least r then | A | ≥ cr and this suﬃces for our purp oses. Step 2 : F or eac h L ∈ 2 ℓ N , let µ L b e the measure on Ω Z d L hc obtained by condi- tioning µ Z d L on the even t { τ f = 1 } , for the element τ ∈ D d L/ℓ whic h maps [0 , ℓ ] d to [0 , ℓ ] d + 1 2 L (1 , . . . , 1) . W e claim that inf L ∈ 2 ℓ N µ L ( f ) > 0 . (62) Indeed, µ L ( f ) = µ L ( f = 1) − µ L ( f = − 1) ≥ 1 − 2 µ L ( f  = 1) = 1 − 2 µ L ( f  = τ f ) ≥ 1 − 2 µ Z d L ( f  = τ f ) µ Z d L ( τ f = 1) = 1 − 4 µ Z d L ( f  = τ f ) 1 − µ Z d L ( f = 0) (63) 27 where the last equality uses that µ Z d L ( f ) = 0 (as ℓ is odd). No w, using (60) with c 0 suﬃcien tly small, we see that that (63) implies (62). The measure µ L ma y b e view ed as a measure on Ω Z d hc supp orted on L Z d -p eriodic conﬁgurations. By compactness, the sequence ( µ L ) L ∈ 2 ℓ N has a weakly con- v erging sub-sequence; denote its limit by µ . The fact that the b o x [0 , ℓ ] d + 1 2 L (1 , . . . , 1) , where the conditioning of µ L is made, tends to inﬁnit y with L implies that µ is a Gibbs measure for the hard-core mo del on Z d . Moreov er, µ ( f ) > 0 due to (62). Ho wev er, we may als o obtain a Gibbs measure µ ′ b y a w eak sub-sequential limit of the measures ( µ Z d L ) L ∈ 2 ℓ N (without conditioning). Since µ ′ ( f ) = 0  = µ ( f ) , there are multiple Gibbs measures. 13.2 Pro of of Lemma 13.1 Let ℓ ∈ N and L ∈ 2 ℓ N . Let λ > 0 , d ∈ N . 13.2.1 W eighted sums Deﬁne a “weigh t function” w : Z d → R supp orted on [0 , ℓ ] d ∩ Z d , by w v : = d Y i =1 ( 1 / 2 v i ∈ { 0 , ℓ } 1 v i ∈ { 1 , . . . ℓ − 1 } . Note that, regardless of the v alue of L , it holds that w − 1 v = | Stab D d L/ℓ ( v ) | where Stab D d L/ℓ ( v ) = { τ ∈ D d L/ℓ : τ v = v } is the stabilizer of the element v (when v is view ed as an element of Z d L ). W e use w to deﬁne a weigh ted sum: for σ ∈ { 0 , 1 } Z d L and A ⊂ Z d L , deﬁne | σ | w A : = X v ∈ A w v σ v . The weigh t function is c hosen so that whenever A ⊂ Z d L is inv ariant under the action of D d L/l , X τ ∈ D d L/l | σ τ | w A = | σ A | . (64) 13.2.2 The lo cal observ able f The observ able f is deﬁned as f : = (1 − 1 B ) · g for the “bad” even t B = { f = 0 } and observ able g that we now deﬁne. Let α : = c α λ 1 + λ ℓ d 28 for a small universal constant c α to b e determined later. Deﬁne B 0 : = { σ : min {| σ | w E , | σ | w O } ≥ α } . Deﬁne F to b e the set of faces of [0 , ℓ ] d , discretized and embedded in Z d L (eac h of the 2 d elements of F is a set of ( ℓ + 1) d − 1 v ertices in Z d L ). F or each H ∈ F deﬁne B H : = { σ : | σ | w H ≤ 2 α } . Deﬁne B : = B 0 ∪ S H ∈F B H and deﬁne g as g : = sgn( | σ | w O − | σ | w E ) where sgn is the sign function. Note that f , g , 1 B , 1 B 0 , 1 B H are [0 , ℓ ] d -lo cal functions that do not dep end on L . 13.2.3 The phases are separated b y con tours Pr o of of item 1 of L emma 13.1. Let s, t b e nearest neigh b ors in Z d L/ℓ . Assume for the sake of contradiction, and without loss of generality , that τ s f = 1 , and τ t f = − 1 . By applying τ − 1 s to σ we may assume without loss of generalit y that f = 1 , τ f = − 1 where τ = τ − 1 s τ t is the reﬂection through some face H ∈ F . W e ha ve σ / ∈ B H ⇐ ⇒ | σ | w H > 2 α, g = 1 , σ / ∈ B 0 = ⇒ | σ | w E = min {| σ | w E , | σ | w O } < α, τ g = − 1 , σ τ / ∈ B 0 = ⇒ | σ τ | w O = min {| σ τ | w E , | σ τ | w O } < α. As vertices of H are ﬁxed by τ , we obtain the contradiction 2 α < | σ | w H = | σ | w E ∩ H + | σ | w O∩ H = | σ | w E ∩ H + | σ τ | w O∩ H < α + α. 13.2.4 The anti-symmetry of f Pr o of of item 2 of L emma 13.1. Consider the reﬂection τ of ( R /L Z ) d , deﬁned b y v = ( v 1 , . . . , v d ) 7→ ( ℓ − v 1 , v 2 , . . . , v d ) , whic h maps [0 , ℓ ] d to itself. While τ / ∈ D d L/ℓ , its action is still deﬁned b y (56). Observ e that τ 1 B = 1 B . How ev er, when ℓ is odd, τ E = O and therefore τ g = − g . Let σ b e sampled from µ Z d L . Finally , as σ and σ τ ha ve the same distribution, we conclude that E [ f ] = E [ τ f ] = E [ − f ] , so that E [ f ] = 0 . 13.2.5 Chessb oard norm b ounds T ow ards establishing item 3 of Lemma 13.1, the next lemma uses Theorem 1.7 (our main result for ﬁnite graphs) to b ound the chessboard norm of the “bad” ev ent B . 29 W e pro ceed to prov e item 3 of Lemma 13.1. F or each A ⊂ Z d L/ℓ , P ( τ s f = 0 ∀ s ∈ A ) = ζ Z d L ( Q s ∈ A τ s 1 B ) Z Z d L  by the chessboard estimate, Proposition 12.1  ≤ ∥ 1 B ∥ | A | ℓ | L ∥ 1 ∥ ( L ℓ ) d −| A | ℓ | L Z Z d L ≤  ∥ 1 B ∥ ℓ | L e − ℓ d 2 ˜ λ  | A | (65) where the in last inequality we used the trivial low er b ound (8), ∥ 1 ∥ ( L ℓ ) d ℓ | L = Z Z d L ≥ e 1 2 ˜ λL d = e ℓ d 2 ˜ λ ( L ℓ ) d . (66) Th us, item 3 of Lemma 13.1 is implied by (65) and the next lemma. Lemma 13.3. Supp ose d ≥ 2 and ℓ = 3 . Ther e exist universal C, c, c α > 0 so that if λ > C log d d then for al l L ∈ 2 ℓ N , ∥ 1 B ∥ ℓ | L ≤ ∥ 1 B ∥ ℓ | 2 ℓ ≤ 5 d exp  1 2 − c λ (1 + λ ) d  ℓ d ˜ λ  . (67) Pr o of. The ﬁrst inequalit y follo ws from Lemma 12.3. F or the second inequality w e consider the mo del on the graph G = Z d 2 ℓ . By the triangle inequality (F act 12.2), ∥ 1 B ∥ ℓ | 2 ℓ ≤ ∥ 1 B 0 ∥ ℓ | 2 ℓ + X H ∈F ,ϵ ∈{± 1 }   1 B H,ϵ   ℓ | 2 ℓ (68) where we deﬁne the even ts B H,ϵ : = B H ∩ { g = ϵ } \ B 0 . W e pro ceed to b ound eac h of these terms. Bounding ∥ 1 B 0 ∥ ℓ | 2 ℓ : Deﬁne the even t ˜ B 0 b y 1 ˜ B 0 : = Q τ ∈ D d 2 ℓ/ℓ τ 1 B 0 . W e ﬁrst relate ˜ B 0 to min {| σ E | , | σ O |} : σ ∈ ˜ B 0 ⇐ ⇒ ^ τ ∈ D d 2 ℓ/ℓ (min {| σ τ | w E , | σ τ | w O } > α ) = ⇒ min ( X τ ∈ D d 2 ℓ/ℓ | σ τ | w E , X τ ∈ D d 2 ℓ/ℓ | σ τ | w O ) > 2 d α  by (64), as E , O are D d 2 ℓ/ℓ -inv ariant  ⇐ ⇒ min {| σ E | , | σ O |} > 2 d α. W e now apply Theorem 1.7 with G = Z d 2 ℓ (whic h is 2 d -regular since ℓ = 3 ) and r = 2 d α = c α λ 1+ λ (2 ℓ ) d . W e verify (15) using that by Lemma 1.10, h ( Z d 2 ℓ ) ≥ c/ℓ 30 and the lo cal expansion prop ert y holds with M LE = c (2 ℓ ) d d and C LE = C . W e also use λ > C log d d and let C here dep end on c α whic h is chosen later. Th us, applying the theorem, we obtain for ℓ = 3 that ∥ 1 B 0 ∥ ℓ | 2 ℓ = ( 2 ℓ ℓ ) d r ζ Z d 2 ℓ  ˜ B 0  ≤ (1 + λ ) 2 − d  | Z d 2 ℓ | 2 − c h ( Z d 2 ℓ ) d r  (69) = exp  ℓ d 2 − c h ( Z d 2 ℓ ) d α  ˜ λ  (70) ≤ exp  1 2 − cc α λ (1 + λ ) d  ℓ d ˜ λ  . (71) Bounding   1 B H,ϵ   ℓ | 2 ℓ : By deﬁnition,   1 B H,ϵ   ℓ | 2 ℓ = ( 2 ℓ ℓ ) d r ζ Z d 2 ℓ  ˜ B H,ϵ  (72) with ˜ B H,ϵ : =      σ : Y τ ∈ D d 2 ℓ/ℓ τ 1 B H,ϵ = 1      . Let σ b e sampled from µ Z d 2 ℓ . By (29), for any K ⊂ Z d 2 ℓ , log ζ Z d 2 ℓ  ˜ B H,ϵ  = I ( σ | ˜ B H,ϵ ) (b y the subadditivity (31) of I) ≤ X v ∈ K I ( σ v | ˜ B H,ϵ ) + X v ∈ K c I ( σ v | ˜ B H,ϵ ) (conca vity of S and Prop osition 3.3) ≤ | K | ( S ( p K ) + p K log λ ) + | K c | ˜ λ (73) where p K : = 1 | K | E h | σ K | | ˜ B H,ϵ i . T o obtain a b ound that will imply (67), we will exhibit a set K whose size is suitably larger than 1 2 | Z d 2 ℓ | on which the o ccupation fraction p K is negligible. W e c ho ose K according to H ∈ F , ϵ ∈ {± 1 } . W e describ e this now for H = { 0 } × { 0 , . . . , ℓ } d − 1 and ϵ = − 1 ; a similar argumen t applies in all other p ossibilities. Set K : =  { 0 } × Z d − 1 2 ℓ  ∪ O = [ τ ∈ D d 2 ℓ/ℓ τ ( H ∪ O ) 31 and note that | K | =  1 2 + 1 4 ℓ  (2 ℓ ) d and | K c | =  1 2 − 1 4 ℓ  (2 ℓ ) d (using that d ≥ 2) . T ow ards b ounding p K , by the deﬁnitions of g , B 0 and B H , σ ∈ { g = ϵ } \ B 0 = ⇒ | σ | w O = min {| σ | w E , | σ | w O } < α, σ ∈ B H ⇐ ⇒ | σ | w H ≤ 2 α. Therefore, σ ∈ ˜ B H,ϵ = ⇒ ^ τ ∈ D d 2 ℓ/ℓ ( | σ τ | w O < α, | σ τ | w H ≤ 2 α ) (since O , H ⊂ K ) = ⇒ ^ τ ∈ D d 2 ℓ/ℓ | σ τ | w K < 3 α = ⇒ X τ ∈ D d 2 ℓ/ℓ | σ τ | w K < 2 d · 3 α  by (64), as K is D d 2 ℓ/ℓ -inv ariant  = ⇒ | σ K | < 2 d · 3 α whic h implies p K < 2 d · 3 α/ | K | ≤ 6 c α λ 1+ λ ≤ λ 1+ λ ≤ min { ˜ λ, 1 } for small c α (recalling ˜ λ = log(1 + λ ) from (18)). Th us S ( p K ) + p K log λ  by monotonicity for p K ≤ λ 1+ λ and (24)  ≤ 6 c α λ 1 + λ log eλ 6 c α λ 1+ λ ≤ C c α λ 1 + λ ( ˜ λ + log C c α )  λ 1 + λ ≤ ˜ λ ≤ 1  ≤ C c α (1 + log C c α ) ˜ λ = : c ′ α ˜ λ (74) and we note that c ′ α c α → 0 − − − − → 0 . Finally , contin ue (73) to obtain log ζ Z d 2 ℓ  ˜ B H,ϵ  ≤ | K | ( S ( p K ) + p K log λ ) + | K c | ˜ λ (b y (74)) ≤ c ′ α | K | ˜ λ + | K c | ˜ λ (substitute | K | , | K c | ) ≤  c ′ α  1 2 + 1 4 ℓ  +  1 2 − 1 4 ℓ  (2 ℓ ) d ˜ λ ≤ (for small c α , and ℓ = 3 ) ≤  1 2 − 1 15  (2 ℓ ) d ˜ λ. (75) End of pro of : Com bining (68), (70), (72) and (75), and ﬁxing c α yields: ∥ 1 B ∥ ℓ | 2 ℓ ≤ exp  1 2 − cc α λ (1 + λ ) d  ℓ d ˜ λ  + 4 d exp  1 2 − 1 15  ℓ d ˜ λ  ≤ 5 d exp  1 2 − c λ (1 + λ ) d  ℓ d ˜ λ  . 32 P art IV Discussion and op en questions 14.1 Connections with earlier w orks Our Part I takes inspiration from [Kah01, Theorem 1.9] [GT04, Proposition 1.5], where it is prov ed that for every ﬁnite simple δ -regular bipartite graph G = ( V , E ) , it holds that Z G ≤ ( Z K d,d ) | V | 2 d . (76) Indeed, some of our pro of steps can b e seen as generalizations of the pro of steps there: Sp eciﬁcally , our deﬁnition of φ is the same as the Q deﬁned in [Kah01]. Equations (3.1) and (3.3) of [Kah01] can then b e view ed as a special case of (45) b y setting λ = 1 , A = { v } and B = N ( v ) , with v a uniformly random element of E . Similarly (3.2) of [Kah01] can b e seen as the ﬁrst inequality of (49). The fact that the entrop y proof of (76) in [GT04] can b e adapted to yield stronger b ounds when applied to suitably restricted subsets of conﬁgurations w as a key p oint in [PS23] (Lemma 4.7 there) and [PS20] (Lemma 5.7 there). This theme is present (in a very diﬀerent form) in our pro of of (41). The use of the free energy functional I in our work is inspired by the (equiv alent) use of Kullback–Leibler divergence in [KS23]. The use of chessboard estimates in Peierls-t ype arguments has b een common since its introduction to the ﬁeld (see [FILS80]). More rare, how ev er, are uses with reﬂection blo c ks of mesoscopic size (so that control of the probabilit y of disseminated bad ev ents b ecomes a signiﬁcant task in its own righ t), and in this w e to ok inspiration from our earlier [HP25]. 14.2 The threshold fugacit y for long-range order on gen- eral graphs Let δ ≥ 2 integer and let G = ( V , E ) denote a ﬁnite simple δ -regular bipartite graph, with bipartition ( E , O ) . Let σ b e sampled from the hard-core measure µ G at fugacity λ . F or which λ do es σ exhibit long-range order? As there is no canonical deﬁnition of long-range order for suc h a ﬁnite graph, we ma y consider, as a proxy , the v alues of λ for which the probability µ G (min {| σ E | , | σ O |} > r ) is small when r is of the order of λ 1+ λ | V | . F or concreteness, let us deﬁne the set of b alanc e d c onﬁgur ations as E Bal := { σ ∈ Ω G hc : min {| σ E | , | σ O |} > 1 10 λ 1 + λ | V |} (77) and consider for which λ do es it hold that µ G ( E Bal ) < 1 10 . (LR O) 33 On the one hand, it is simple to see that (LRO) fails when λ ≤ c 0 δ with c 0 > 0 a suﬃciently small universal constant (as long as λ 1+ λ | V | ≥ C , where, as usual, C > 0 is a suﬃciently large universal constant). On the other hand, Theorem 1.7 shows that (LR O) holds for all λ > C log δ δ when G satisﬁes mild expansion properties (global expansion h ( G ) ≥ c and lo cal expansion with parameters C LE ≤ C and M LE ≥ cδ log δ suﬃces, as long as λ 2 δ | V | ≥ C ). Th us, with mild assumptions on G , the threshold fugacity for long-range order (in the sense of (LRO)) lies b et ween order 1 δ and order log δ δ . Conjecture 1.2 states that the threshold fugacit y has order 1 δ for the (inﬁnite) lattice Z d , and w e exp ect the same to b e true for the (ﬁnite) h yp ercub e graph. It is tempting to think that the threshold fugacit y should alwa ys be of order 1 δ under such assumptions. How ever, we now show with an example that the threshold ma y ha ve order log δ δ , sho wing that the b ound resulting from our main result, Theo- rem 1.7, cannot b e improv ed without further assumptions. Example 14.4. First, we deﬁne a “linear gadget” graph: The linear gadget with a single blo c k is and the linear gadget with tw o blo c ks is . Similarly , for integer m ≥ 1 , w e let L m b e the linear gadget with m blo c ks. The imp ortant features of L m are that it is bipartite, the endp oin t v ertices ℓ, r in hav e degree 1 , all other vertices hav e degree 3 and its threshold fugacity for long-range order is of order m in the following sense: F or m ≥ C , at fugacity at most cm , we hav e µ L m  min {| σ E | , | σ O |} > 1 5 λ 1 + λ | V ( L m )     σ ℓ , σ r  ≥ 99 100 (78) almost surely (on the o ccupancy status of the endp oint vertices ℓ, r ). Second, we deﬁne a “stretching op eration” by the linear gadget: F or integer m ≥ 1 and a 3 -regular graph H , let ¯ H m b e the graph formed from H by replacing edge { u, v } of H with the linear gadget L m , with u, v identiﬁed with the endp oin ts ℓ, r of L m . Note that ¯ H m is also 3 -regular, and that it is bipartite when H is. One may chec k that the Cheeger constant satisﬁes h ( ¯ H m ) = Θ( h ( H ) /m ) (i.e., the ratio of the tw o sides is b et ween t wo p ositiv e universal constants). Third, for integers m ≥ 1 , d ≥ 2 and a d -regular graph F , we let F m b e the m - blo w-up of F . That is, the graph with V ( F m ) := { ( v , i ) : v ∈ V ( F ) , 1 ≤ i ≤ m } and ( v , i ) adjacent to ( u, j ) if { u, v } ∈ E ( F ) . Note that F m is a dm -regular graph. One may c heck that the Cheeger constant satisﬁes h ( F m ) = Θ( mh ( F )) , and also that F m satisﬁes lo cal expansion with parameters C LE = 1 2 d 2 and M LE = | E ( F ) | (the latter fact follows by letting ( i v ) v ∈ F uniform in { 1 , . . . , m } indep enden tly and taking the connected graph T to b e the induced graph on { ( v , i v ) : v ∈ F } ). Finally , we deﬁne our example graph: Let δ b e a large multiple of 3 and let H b e a 3 -regular bipartite expander graph (of arbitrary size), i.e., having Cheeger 34 constan t h ( H ) ≥ c for a universal c > 0 . Our example graph is G := ( ¯ H δ ) δ / 3 (the δ / 3 -blow-up of the stretching of H by the linear gadget L δ ). Observe that G is a δ -regular bipartite graph with Cheeger constan t h ( G ) ≥ c and lo cal expansion with parameters C LE = 9 2 and M LE = 9 2 | V ( G ) | δ . Ho wev er, (LRO) fails for G at fugacities λ ≤ c 0 log δ δ for a suﬃcien tly small c 0 > 0 . Indeed, the hard-core mo del on G is equiv alen t to the hard-core mo del on ¯ H δ (declaring a v ertex v acant if and only if all its copies in G are v acant) with fugacity λ ′ = (1 + λ ) δ / 3 − 1 ≤ δ c 0 / 3 ≪ δ , whence the claim follows from the prop ert y (78) of the linear gadget. 14.3 The probabilit y of a balanced conﬁguration on the h yp ercub e W e contin ue with the terminology of balanced conﬁgurations of the previous section (see (77)). Theorem 1.7 b ounds the probability of a balanced conﬁgura- tion for suﬃciently large fugacities. How sharp is this b ound? W e discuss this question for the hypercub e graph G = Z d 2 . W e present tw o natural low er b ounds on the probability of a balanced conﬁgu- ration: 1. Iden tify the vertex set of the hypercub e Z d 2 with { 0 , 1 } d and denote the Hamming weigh t of v ∈ Z d 2 b y | v | = P d i =1 v i . Let E Ham := { σ ∈ Ω Z d 2 hc : σ v = 0 for v ∈ E with | v | ≤ d 2 or v ∈ O with | v | > d 2 } b e the set of hard-core conﬁgurations which hav e solely o dd o ccupation for v ertices of Hamming w eight at most d 2 and solely even o ccupation for Hamming weigh t at v ertices of Hamming weigh t more than d 2 . It is simple to chec k that ζ Z d 2 ( E Ham ) ≥ (1 + λ ) ( 1 2 − C √ d )2 d . It is not hard to see that conditioning the hard-core mo del on Z d 2 to b e in E Ham t ypically yields a balanced conﬁguration (sa y , with probability at least 1 2 ) when λ ≫ 1 2 d . Now, using Corollary 1 . 4 w e arrive at the following lo wer b ound µ Z d 2 ( E Bal ) ≥ 1 2 ζ Z d 2 ( E Ham ) ζ Z d 2 (Ω Z d 2 hc ) ≥ c (1 + λ ) − C √ d 2 d , λ ≥ C log d d . (79) 2. A diﬀerent low er b ound is obtained by considering the set E coord := { σ ∈ Ω Z d 2 hc : σ v = 0 for v ∈ E with v 1 = 0 or v ∈ O with v 1 = 1 } of hard-core conﬁgurations which hav e solely o dd o ccupation at vertices with ﬁrst co ordinate 0 solely ev en o ccupation at v ertices with ﬁrst co or- 35 dinate 1 . Here we hav e ζ Z d 2 ( E coord ) = (1 + 2 λ ) 1 4 2 d = (1 + λ ) 1 2 2 d  1 + λ 2 1 + 2 λ  − 1 4 2 d . Again, conditioning the hard-core mo del on Z d 2 to b e in E coord t ypically yields a balanced conﬁguration (say , with probabilit y at least 1 2 ) when λ ≫ 1 2 d . Therefore, with Corollary 1 . 4 , we arrive at the low er b ound µ Z d 2 ( E Bal ) ≥ 1 2 ζ Z d 2 ( E coord ) ζ Z d 2 (Ω Z d 2 hc ) ≥ c  1 + λ 2 1 + 2 λ  − 1 4 2 d , λ ≥ C log d d . (80) W e note that this lo wer b ound impro ves up on (79) in the regime C log d d ≤ λ ≤ c √ d . W e b eliev e that these scenarios capture the leading b eha vior of the (log of the) probabilit y of the balanced even t. Moreov er, though w e ha ve not shown the corresp onding lo wer bound, this may hold also in the wider range λ ≥ C d . Thus, w e arrive at Conjecture 14.5. Ther e exist C , d 0 > 0 such that for al l dimensions d ≥ d 0 , µ Z d 2 ( E Bal ) =    e − Θ( λ 2 2 d ) C d ≤ λ ≤ 1 √ d e − Θ( log(1+ λ ) √ d 2 d ) λ ≥ 1 √ d (81) with the Θ notation signifying that the b ound is tight up to universal multiplic a- tive c onstants. Our results (Corollary 1.3) provide an upper b ound on µ Z d 2 ( E Bal ) in the regime λ ≥ C log d d , which captures the b eha vior up to a p o wer of d in the exp onen t. 14.4 P erio dic Gibbs measures and sim ultaneous p ercola- tion As brieﬂy mentioned in Section 1.2, the hard-core mo del on Z d admits inﬁnite- v olume limits with even/odd b oundary conditions, yielding t wo extremal Gibbs measures µ even , µ odd , whic h are also in v ariant to the parit y-preserving automor- phisms of Z d . Moreo ver, m ultiple Gibbs measures exist if and only if these t wo measures are distinct, so that our results show that µ even  = µ odd when λ > C log d d . It is natural to ask whether, for a given fugacity λ > 0 , every p eriodic (i.e., in v ariant to a full rank lattice of translations) Gibbs measure is a mixture of µ even and µ odd . It may well b e the case that this prop ert y holds for all λ > 0 (an analogous prop ert y is known to hold at all temp eratures in the Ising mo del; see [Rao20] and references therein). How ever, our results do not directly imply this, even in the range λ > C log d d , and the current state-of-the art is that the 36 prop ert y is known only when λ > C log 3 / 2 d d 1 / 4 [PS20, Section 3.1.1] (or when the Gibbs meas ure is unique). Related to this is the notion of simultane ous p er c olation : When sampling a conﬁguration σ from a Gibbs measure µ , do es an inﬁnite connected comp onen t of o dd-o ccupied v ertices and an inﬁnite connected comp onen t of even-occupied v ertices exist simultaneously , where connectivit y is measured at graph distance 2 ? On the one hand, almost sure absence of simultaneous p ercolation for a p e- rio dic µ easily implies that µ is a mixture of µ even or µ odd . On the other hand, o ccurrence of sim ultaneous percolation is an obstacle to direct uses of the P eierls argumen t or cluster expansion techniques (see also [JP20, b elo w Theorem 6]). In high dimensions, σ | E (or σ | O ) sto c hastically dominates Bernoulli percolation (on E with connectivity at distance 2) of parameter λ λ +(1+ λ ) 2 d , whence almost sure simultaneous p ercolation o ccurs when C d 2 < λ < c log d d (as for minority p er- c olation in the Ising mo del [ABL87]). W e exp ect that there is no simultaneous p ercolation when λ > C log d d . 15 A c kno wledgmen ts W e are grateful to W o jtek Samotij for initial discussions of the problem and the approach. W e thank Gady Kozma, W o jtek Samotij and Yinon Spink a for earlier collab orations on this and related problems, and Michal Bassan, Quen tin Dubroﬀ, Jeﬀ Kahn, Jonatan Hadas, Marcus Mic helen, Jin young P ark and Dana Randall for several useful discussions. Supp ort from the National Science F oundation grant DMS-2451133, Europ ean Researc h Council Consolidator gran t 101002733 (T ransitions) and Israel Science F oundation grants 2110/22 and 2340/23 is gratefully ackno wledged. References [ABL87] Mic hael Aizenman, Jean Bricmont, and Jo el L. Leb owitz. Percolation of the minority spins in high-dimensional Ising models. Journal of Statistic al Physics , 49(3):859–865, 1987. [AS08] Noga Alon and Joel H. Sp encer. The Pr ob abilistic Metho d , chapter 15, pages 266–269. Wiley-Interscience, 2008. [BHW99] Graham R. Brigh tw ell, Olle Häggström, and Peter Winkler. Non- monotonic Behavior in Hard-Core and Widom–Rowlinson Mo dels. Journal of Statistic al Physics , 94(3):415–435, F ebruary 1999. [Bis09] Marek Biskup. Reﬂection p ositivit y and phase transitions in lattice spin models. In Metho ds of Contemp or ary Mathematic al Statistic al Physics , pages 1–86. Springer, 2009. [BM94] J. V an Den Berg and C. Maes. Disagreemen t p ercolation in the study of Marko v ﬁelds. The Annals of Pr ob ability , 22(2):749–763, April 1994. 37 [CGFS86] F an RK Chung, Ronald L Graham, P eter F rankl, and James B Shearer. Some intersection theorems for ordered sets and graphs. Jour- nal of Combinatorial The ory, Series A , 43(1):23–37, 1986. [Cha15] Y ao-Ban Chan. Upp er bounds on the growth rates of hard squares and related mo dels via corner transfer matrices. Discr ete Mathematics & The or etic al Computer Scienc e , DMTCS Pro ceedings, 27th Interna- tional Conference on F ormal P ow er Series and Algebraic Combinatorics (FPSA C 2015)(Pro ceedings), January 2015. [CHV20] Endre Csók a, Viktor Harangi, and Bálint Virág. Entrop y and expan- sion. Annales de l’Institut Henri Poinc ar é, Pr ob abilités et Statistiques , 56(4), Nov ember 2020. [CT98] F an RK Chung and Prasad T etali. Isop erimetric inequalities for carte- sian pro ducts of graphs. Combinatorics, Pr ob ability and Computing , 7(2):141–148, 1998. [CT05] Thomas M. Cov er and Joy A. Thomas. Elements of Information The- ory , chapter 2. John Wiley & Sons, Ltd, 2005. [CW98] Neil J. Calkin and Herb ert S. Wilf. The Number of Indep enden t Sets in a Grid Graph. SIAM Journal on Discr ete Mathematics , 11(1):54–60, F ebruary 1998. [DG13] Margherita Disertori and Alessandro Giuliani. The nematic phase of a system of long hard ro ds. Communic ations in Mathematic al Physics , 323(1):143–175, 2013. [Dob68a] Roland L’vo vic h Dobrushin. The description of a random ﬁeld by means of conditional probabilities and conditions of its regularit y . The- ory of Pr ob ability & Its Applic ations , 13(2):197–224, 1968. [Dob68b] Roland L’v ovic h Dobrushin. The problem of uniqueness of a Gibbsian random ﬁeld and the problem of phase transitions. F unctional A nalysis and its Applic ations , 2(4):302–312, 1968. [FILS78] Jürg F röhlich, Rob ert Israel, Elliott H. Lieb, and Barry Simon. Phase transitions and reﬂection p ositivit y . I. General theory and long range lattice mo dels. Communic ations in Mathematic al Physics , 62(1):1–34, Jan uary 1978. [FILS80] Jürg F röhlic h, Rob ert B. Israel, Elliott H. Lieb, and Barry Simon. Phase transitions and reﬂection p ositivity . I I. Lattice systems with short-range and Coulomb interactions. Journal of Statistic al Physics , 22(3):297–347, March 1980. [FV17] Sacha F riedli and Y v an V elenik. Statistic al Me chanics of L attic e Sys- tems: A Concr ete Mathematic al Intr o duction . Cambridge Universit y Press, Cambridge, 2017. 38 [Gal11] David Galvin. A threshold phenomenon for random indep endent sets in the discrete hypercub e. Combinatorics, pr ob ability and c omputing , 20(1):27–51, 2011. [Gal19] David Galvin. Indep enden t sets in the discrete hypercub e. arXiv pr eprint arXiv:1901.01991 , 2019. [Geo11] Hans-Otto Georgii. Gibbs Me asur es and Phase T r ansitions . De Gruyter, 2011. [GK04] David Galvin and Jeﬀ Kahn. On phase transition in the hard-core mo del on Z d . Combinatorics, Pr ob ability and Computing , 13(2):137– 164, 2004. [GT04] Da vid Galvin and Prasad T etali. On weigh ted graph homomorphisms. DIMA CS Series in Discr ete Mathematics and The or etic al Computer Scienc e , 63:97–104, 2004. [Had22] Daniel Hadas. A minimal introduction to the c hessboard estimate, 2022. http://daniel.hadaso.net/chessboard.pdf . [HJ24] Qidong He and Ian Jauslin. High-fugacit y expansion and crystalliza- tion in non-sliding hard-core lattice particle mo dels without a tiling constrain t. Journal of Statistic al Physics , 191(10):135, 2024. [Ho e63] W assily Ho eﬀding. Probabilit y Inequalities for Sums of Bounded Random V ariables. Journal of the Americ an Statistic al Asso ciation , 58(301):13–30, 1963. [HP25] Daniel Hadas and Ron Peled. Columnar order in random packings of 2 × 2 squares on the square lattice. Journal of the Eur op e an Mathe- matic al So ciety , 2025. [JK23] Matthew Jenssen and Peter Keev ash. Homomorphisms from the torus. A dvanc es in Mathematics , 430:109212, 2023. [JMP26] Matthew Jenssen, Alexandru Malekshahian, and Jiny oung Park. A reﬁned graph container lemma and applications to the hard-core mo del on bipartite expanders. R andom Structur es & Algorithms , 68(1):e70041, 2026. [JP20] Matthew Jenssen and Will Perkins. Indep enden t sets in the hypercub e revisited. Journal of the L ondon Mathematic al So ciety , 102(2):645–669, Octob er 2020. [Kah01] Jeﬀ Kahn. An En tropy Approac h to the Hard-Core Model on Bipartite Graphs. Combinatorics, Pr ob ability and Computing , 10(3):219–237, Ma y 2001. [KL99] Jeﬀ Kahn and Alexander Lawrenz. Generalized rank functions and an en tropy argumen t. Journal of Combinatorial The ory, Series A , 87(2):398–403, 1999. 39 [KS83] A. D. Korsh unov and A. A. Sap ozhenk o. The n umber of binary co des with distance 2. Pr oblemy Kib ernet. , 40:111–130, 1983. [KS23] Gady Kozma and W o jciec h Samotij. Low er tails via relativ e en tropy . The Annals of Pr ob ability , 51(2):665–698, 2023. [Lo v15] Shachar Lo vett. Information theory in combinatorics (notes), 2015. https://simons.berkeley.edu/talks/ information- theory- combinatorics- i . [Lo v23] László Lovász. Submo dular setfunctions on sigma-algebras, version 2. arXiv pr eprint arXiv:2302.04704 , 2023. [P el22] Ron Peled. Random packings and liquid crystals (online talk), 2022. https://www.youtube.com/watch?v=oKmfKBHZgBI . [PS14] Ron Peled and W o jciec h Samotij. Odd cutsets and the hard-core mo del on Z d . In Annales de l’IHP Pr ob abilités et Statistiques , v olume 50, pages 975–998, 2014. [PS19] Ron Peled and Yinon Spink a. Lectures on the spin and loop O(n) mo dels. Sojourns in Pr ob ability The ory and Statistic al Physics - I , 2019. [PS20] Ron Peled and Yinon Spink a. Long-range order in dis crete spin sys- tems. arXiv pr eprint arXiv:2010.03177 , 2020. [PS23] Ron Peled and Yinon Spink a. Rigidity of prop er colorings of Z d . In- ventiones mathematic ae , 232(1):79–162, 2023. [Rao20] Aran Raouﬁ. T ranslation-inv ariant Gibbs states of the Ising mo del: General setting. The Annals of Pr ob ability , 48(2):760–777, 2020. [R Y10] An up Rao and Jijiang Y an. CSE533: Information theory in computer science, 2010. https://homes.cs.washington.edu/~anuprao/pubs/ CSE533Autumn2010/lecture7.pdf . [Shl86] SB Shlosman. The metho d of reﬂection positivity in the mathematical theory of ﬁrst-order phase transitions. R ussian Mathematic al Surveys , 41(3):83–134, 1986. [Til00] Jean-Pierre Tillic h. Edge isop erimetric inequalities for pro duct graphs. Discr ete Mathematics , 213(1-3):291–320, 2000. [Tim12] Ádám Timár. Boundary-connectivity via graph theory . Pr o c e e dings of the Americ an Mathematic al So ciety , 141(2):475–480, June 2012. [vS94] J. v an den Berg and J.E. Steif. Percolation and the hard-core lattice gas model. Sto chastic Pr o c esses and their Applic ations , 49(2):179–197, 1994. [W ei06] Dror W eitz. Coun ting indep enden t sets up to the tree threshold. In Pr o c e e dings of the thirty-eighth annual ACM symp osium on The ory of c omputing , pages 140–149, 2006. 40 [Wik25] Wikip edia con tributors. Hamming co de — Wikip edia, the free encyclo- p edia. https://en.wikipedia.org/w/index.php?title=Hamming_ code&oldid=1319758292 , 2025. [Online; accessed 11-Nov ember-2025]. A Lo cal expansion via random walks In this section we prov e Lemma 1.6, which b ounds the lo cal expansion param- eters of a graph using prop erties of simple random walk on that graph. Let G = ( V , E ) b e a ﬁnite δ -regular bipartite graph. Let C 0 ≥ 1 and let M 0 ≥ 1 in teger. Introduce the (ﬁnite length) Green’s function g : V × V → [0 , ∞ ) : g u,w = E [ |{ 0 ≤ i < M 0 : W i = w }| ] with W 0 , . . . , W M 0 − 1 a simple random walk on G starting at W 0 = u . Lemma 1.6 supp oses that for each v ∈ V , g v ,v − 1 ≤ C 0 − 1 δ . (82) W e use the follo wing simple consequence of the symmetry of the transition matrix: Lemma A.1. L et u  = w b e vertic es in the same bip artite class of G . Then g u,w ≤ q ( g u,u − 1)( g w,w − 1) . Pr o of. Let P b e the transition matrix for the simple random on G . Denote M : = P 0 0] ≤ sup u ∈ N ( v ) E [ n v | W 0 = u ] = sup u ∈ N ( v ) X w ∈ N ( v ) g u,w ( b y Lemma A.1 ) ≤ sup u ∈ N ( v ) h g u,u + p g u,u − 1 X w ∈ N ( v ) \{ u } p g w,w − 1 i ( b y (82) ) ≤ C 0 . Therefore P ( W ∩ N ( v )  = ∅ ) = P ( n v > 0) = E [ n v ] E [ n v | n v > 0] ≥ M 0 C 0 δ | V | . Let W 0 , . . . , W M 0 − 1 b e as in Lemma A.2. T o ﬁnish the pro of of Lemma 1.6 it remains to v erify that the graph T deﬁned as the graph spanned by the edges of the w alk W satisﬁes the prop erties in the deﬁnition of lo cal expansion (Deﬁnition 1.5) with parameters C LE = C 0 and M LE = M 0 . The fact that each W i is distributed uniformly in V implies, b y the union b ound, that P ( uv ∈ E ( T )) ≤ M 0 | E | for eac h uv ∈ E , while the fact that P ( N ( u ) ∩ V ( T )  = ∅ ) ≥ M 0 C 0 δ | V | for each u ∈ V is exactly (83). B Pro ofs of generalized Shearer’s inequality The generalized Shearer’s inequality (Prop osition 3.1) has app eared in sev eral places, in form ulations that slightly diﬀer from ours, see [R Y10, Lemma 1], [CHV20, Theorem A.2] and [Lov15, Lemma 2.2]. W e provide here tw o pro ofs for Prop osition 3.1. Pr o of. F or each ﬁxed K ⊂ J , denote 𭟋 ( K ) : = S (( X j ) j ∈ K ) . Then 𭟋 is a sub- mo dular set-function: 𭟋 ( K 1 ∩ K 2 ) + 𭟋 ( K 1 ∩ K 2 ) ≤ 𭟋 ( K 1 ) + 𭟋 ( K 2 ) . F or g : J → [0 , ∞ ) deﬁne the functional ˆ 𭟋 by the Cho quet in tegral: ˆ 𭟋 ( g ) = Z ∞ 0 𭟋 ( { j ∈ J : g ( j ) ≥ t } ) dt. Then b y [Lov23, Corollary 4.10] if follows that ˆ 𭟋 is a conv ex functional. By Jensen’s in equalit y it holds that pS ( X ) = ˆ 𭟋 ( p 1 J ) ≤ E ˆ 𭟋 ( 1 K ) = S (( X j ) j ∈ K | K ) . 42 Pr o of. W rite X

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