Distribution of the magnetization of the critical Ising model on sparse random graphs

Distribution of the magnetization of the critical Ising mo del on sparse random graphs Kyprianos-Iason Pro dromidis ∗ Allan Sly † Abstract In this pap er, w e consider the Ising mo del on random d -regular graphs (with d ≥ 3 ) and Erdös-Rén yi graphs G ( n, d/n ) (with d > 1 ) at the critical temp erature. W e pro ve that the magnetization , i.e. the sum of the spins of a conﬁguration, is t ypically of order n 3 / 4 and when m ultiplied by n − 3 / 4 con verges in distribution to a non-trivial random v ariable, whose density w e describ e. In the regular graph case, the Small Subgraph Conditioning Method applies, and the limiting densit y is of the form 1 Z exp( − C d z 4 ) . Surprisingly , in the Erdös-Rényi case, while the ratio of the second momen t and ﬁrst momen t squared is bounded, the short cycle coun t is not enough to explain the ﬂuc- tuations of the partition function restricted to a particular magnetization. W e identify the additional source of randomness as path coun ts of slowly diverging length. This quan tity is motiv ated b y the heuristic that correlations b et ween distant vertices are prop ortional to their lo cal branc hing rate. Augmenting the Small Subgraph Condition- ing Metho d with these path coun ts allo ws us to pro v e con vergence of the magnetization to a non-deterministic limiting distribution. T o our knowledge, the need to condition on graph observ ables beyond the cycle counts is a new phenomenon for spin systems. As further corollaries, we deriv e a p olynomial lo w er b ound on the mixing time of the stochastic Ising model on sparse random graphs at the critical temperature complemen ting recent upper b ounds [3, 21]. Moreov er, we establish the ﬂuctuations of the free energy in the Erdös-Rényi case, answ ering a recent question of Co ja-Oghlan et. al. [5]. 1 In tro duction Just as for the lattice Z d , on sparse random graphs the Ising mo del exhibits a phase transition with exp onential deca y of correlation at high temp eratures and long-range order emerging at low temperatures. Because these graphs are lo cally treelike, man y prop erties of the Ising mo del can b e understoo d in terms of the Ising mo del on its lo cal w eak limit graph. In the lo w temp erature regime, the lo cal weak limit of the Ising mo del and graph together w as sho wn to b e a balanced mixture of the extremal plus and minus measures on the lo cal weak limit tree (either the inﬁnite d -regular tree for random d -regular graphs or the Galton-W atson ∗ Departmen t of Mathematics, Princeton Universit y , Email: kp2702@princeton.edu † Departmen t of Mathematics, Princeton Universit y , Email: allansly@princeton.edu 1 branc hing pro cess with Poisson oﬀspring distribution on G ( n, d/n ) ) and the free energy is explicitly known [2, 6, 11, 19]. W e are interested in the law of the magnetization. The simplest graph to study the Ising mo del is the me an-ﬁeld setting, where the underlying graph is the complete graph. In this case, if n is the n umber of v ertices and the inv erse temperature is β /n , the mo del undergoes a similar phase transition at β = 1 (see for example [9]). When β < 1 , the magnetization m n = n P i =1 σ i ob eys a central limit theorem with m n · n − 1 / 2 ( d ) − − − → n →∞ N (0 , (1 − β ) − 1 ) . On the other hand, when β > 1 , the distribution is bimo dal: 1 n m n concen trates near ± x , where x solves the equation x = tanh( β x ) . Finally , in the critical case β = 1 , m n is of order n 3 / 4 and m n · n − 3 / 4 con verges in distribution to a la w with density ∝ exp( − y 4 / 12) (see [12]). In the critical windo w when β = 1 + θ n − 1 / 2 , for some θ ∈ R , the limiting density b ecomes ∝ exp( θ y 2 / 2 − y 4 / 12) . The case of the distribution of the magnetization for the critical Ising mo del in Z d remains unresolv ed. In dimensions 5 and higher it is natural to exp ect b eha viour similar to the mean ﬁeld case. There is, ho wev er, a subtlet y in the eﬀect of the b oundary conditions. In a b ox with free b oundary conditions the magnetization is in fact Gaussian. This is b ecause the missing edges on the b oundary sligh tly lo w er the av erage degree and eﬀectiv ely push the system in to the high temp erature regime. This is not an issue with p erio dic b oundary conditions in whic h case [17] show ed that there is a non-normal magnetization and conjectured that it has asymptotic densit y exp( − cy 4 ) as in the complete graph. The case of d = 2 is quite diﬀeren t, as the limiting distribution was established in [ 4] and has tails exp( − ( c + o (1)) x 16 ) . In this pap er, we study the same problem at criticalit y on sparse random graphs, more sp eciﬁcally on random d -regular graphs (for d ≥ 3 ) and Erdös-Rényi graphs G ( n, d/n ) (for d > 1 ). In both cases, at the critical temperature w e ﬁnd that the limiting law of the magnetization is non-Gaussian. T o describ e conv ergence to a random measure, we will consider conv ergence in the W asser- stein distance with resp ect to a K olmogoro v–Smirnov metric. Sp eciﬁcally , on the space M 1 ( R ) of probability measures on R , consider the metric d KS ( µ, ν ) := sup y ∈ R | µ (( −∞ , y ]) − ν (( −∞ , y ]) | . W e are in terested in the 1-W asserstein distance asso ciated to the d KS metric. In the random regular case the limit is, in fact, deterministic, and the same as for the critical Curie-W eiss mo del, up to a d -dep endent constant in the exp onent. Theorem 1.1. Consider the Ising mo del on a gr aph G ∼ G n,d (for d ≥ 3 ) at the critic al temp er atur e, β c = tanh − 1 (( d − 1) − 1 ) . L et µ n, r e g ∈ M 1 ( R ) b e the r andom me asur e deﬁne d as µ n, r e g ( A ) := P n, r e g ( m · n − 3 / 4 ∈ A | G ) , with G ∼ G n,d . Then, ther e exists a me asur e µ r e g ∈ M 1 ( R ) such that µ n, r e g − − − → n →∞ µ r e g in the 1-W asserstein distanc e asso ciate d to d KS . This me asur e µ r e g has density f r e g ( y ) ∝ exp  − ( d − 2)( d − 1) 12 d 2 y 4  . 2 The case of the Erdós-Rënyi random graph is more complicated; the limiting distribution is not deterministic as in the random regular case. The limit tak es the form of a mixture of the distributions found in the critical windo w for the Curie-W eiss mo del. Theorem 1.2. Consider the Ising mo del on a gr aph G ∼ G ( n, d/n ) (for d > 1 ) at the critic al temp er atur e, β c = tanh − 1 ( d − 1 ) . L et µ n ∈ M 1 ( R ) b e the r andom me asur e deﬁne d as µ n ( A ) := P ( m · n − 3 / 4 ∈ A | G ) , with G ∼ G ( n, d/n ) . Then, ther e exists a r andom me asur e µ ∈ M 1 ( R ) such that µ n − − − → n →∞ µ in the W asserstein distanc e asso ciate d to d KS . Mor e over, the me asur e µ is given by µ = µ ( X ) , wher e for x ∈ R , µ ( x ) ( A ) = R A exp  x √ 2( d − 1) · u 2 −  1 4( d − 1) + 1 12  · u 4  d u R R exp  x √ 2( d − 1) · u 2 −  1 4( d − 1) + 1 12  · u 4  d u , and X ∼ N (0 , 1) . The natural approach for studying the magnetization of spin systems on random graphs is to apply the second momen t metho d to the partition function and its restriction to a particular magnetization. One commonly ﬁnds that E ( Z 2 n ) E ( Z n ) 2 − − − → n →∞ c ∈ (1 , ∞ ) implying that the normalized partition function retains some non-trivial asymptotic v ariance. This v ariance is typically explained b y the eﬀect of small cycles in the graph as was ﬁrst demonstrated b y Robinson and W ormald in the study of Hamiltonicit y of random regular graphs [22] where they in tro duced the Small Subgraph Conditioning Metho d. This metho d has b een applied to a range of spin systems such as the ferromagnetic (e.g. [5]) and anti- ferromagnetic (e.g. [10]) Ising model, the hardcore mo del (e.g. [20]) and colorings (e.g. [8, 16]). The heuristic explanation for the small graph conditioning metho d is that adding an edge of a graph that closes a cycle has a diﬀerent eﬀect on the partition function than adding an edge b etw een distan t vertices; in the uniqueness regime, nearby vertices are correlated while distan t ones are almost uncorrelated. F or each i , the num b er of cycles of length i in the graph, Y i,n , is asymptotically P oisson with mean λ i with a jointly indep enden t limit. F or appropriately c hosen α i , κ i and a slowly gro wing m n , one can apply the second moment metho d to Z n m n Y i =3 exp( − α i C i + κ i ) and show that it is asymptotically constan t. Consequently , for C i indep enden t P oisson ( λ i ) , Z n E ( Z n ) ( d ) − − − → n →∞ W = exp ∞ X i =1 ( α i C i − κ i ) ! 3 The same metho d can b e applied to the partition function, restricted to a giv en magnetiza- tion, which in turn yields the asymptotic law of the magnetization. This metho d has b een previously applied for th e high-temp erature and the anti-ferromagnetic regime up to the reconstruction threshold. It is, therefore, reasonable to assume that it should also apply at the critical temp erature where decay of correlation still holds and indeed it do es for the regular graph case. In the case of the Erdös-Rényi random graph, it is necessary to accoun t for the ﬂuctuations in the total num b er of edges. This can b e done in t wo w a ys: one is to ﬁx the num b er of edges (whic h is the route tak en in [5]) and the second is to reweigh t the partition function by a factor of (cosh( β )) −| E | , as eac h extra edge increases the partition function b y approximately a cosh( β ) factor. After either of these adjustments, in the high temp erature regime, the Small Subgraph Conditioning Metho d w orks as normal. How ev er, at the critical temp erature, even after adjusting for the edge coun t and the eﬀect of small cycles, a non-trivial amoun t of v ariance remains asymptotically . In fact, when d ∈ (1 , 4) , E [( Z n (cosh( β )) −| E n | ) 2 ] E [( Z n (cosh( β )) −| E n | ] 2 → ∞ . Ho wev er, if w e only consider conﬁgurations whose magnetizations are m , i.e. w e set Z n,m = X σ : P v σ v = m exp β X u ∼ v σ u σ v ! and assume that m · n − 3 / 4 → x then E [( Z n,m (cosh( β )) −| E n | ) 2 ] E [( Z n,m (cosh( β )) −| E n | ] 2 → c ( x, d ) ∈ (1 , ∞ ) . T o understand the source of this surprising extra v ariance, consider the pro cess of building the graph one edge at a time with G i the graph after i edges. When adding the i + 1 edge ( u, v ) , the c hange in the partition function is given b y Z G i +1 Z G i = cosh( β ) (1 + tanh( β ) · E G i ( σ u σ v )) Note that if the edge closes a small cycle, it means that u and v w ere close in G i and there- fore E G i ( σ u σ v ) ≈ (tanh( β )) d ( u,v ) . This is the cycle eﬀect in the Small Subgraph Conditioning Metho d. T w o random vertices will typically hav e E G i ( σ u σ v ) ≍ n − 1 2 at the critical temp era- ture (at least for the ﬁnal edges of the construction). Moreo ver, ho w correlated σ u and σ v are will dep end on how well u and v are connected to the gian t comp onent of the graph. Heuristically , from a consideration of the critical FK-comp onent, w e exp ect E G i ( σ u σ v ) ≍ n − 1 2 d − 2 ℓ S ℓ ( u ) S ℓ ( v ) , (1) where S ℓ ( u ) is the n umber of vertices at distance ℓ from u . This suggests that the v ariance of E G i ( σ u σ v ) is of order C /n and so summed ov er order n edges these weak correlations should induce order one m ultiplicativ e ﬂuctuations in the partition function whic h explains the failure of the v anilla implemen tation of the Small Subgraph Conditioning Metho d. 4 While w e won’t need to pro ve (1), it suggests the right adjustment. If the cov ariance is prop ortional to E G i ( σ u σ v ) then it is also appro ximately prop ortional to the n umber of additional paths of length 2 ℓ + 1 created when adding the edge ( u, v ) . This appro ximation is more accurate for larger v alues of ℓ . W e let X ℓ,n denote the num b er of paths of length ℓ presen t in G and b X ℓ,n := X ℓ,n − 1 2 nd ℓ q 1 2 nd 2 ℓ ℓ 2 d − 1 . (2) While we don’t prov e it, as n → ∞ , the ( b X ℓ,n ) ℓ ≥ 3 con verges join tly in distribution to a join tly Gaussian ensemble of normal random v ariables ( X ∗ ℓ ) ℓ ≥ 3 . Moreo v er, that X ∗ ℓ con verge in probabilit y as ℓ → ∞ to some standard normal random v ariable X ∗ ∞ . This quan tit y gives a measure of the amoun t of long paths in the graph and explains the additional source of the v ariance. As suc h, our approach is to apply the Small Subgraph Conditioning Metho d with X ℓ n ,n in addition to the cycle counts for some slo wly diverging ℓ n . As far as w e kno w, this is the ﬁrst example of a spin system where augmen ting the cycles count with an additional observ able of the graph is necessary to apply the Small Subgraph Conditioning Metho d (see [23] in the case of Hamiltonian cycles with randomly selected oriented edges requires extra observ ables from the edge decorations). W e note that Theorems 1.1 and 1.2 also imply low er bounds for the mixing times of the critical sto chastic Ising mo del for both regular and Erdös-Rén yi graphs. This is the ﬁrst p olynomial lo w er b ound for this c hain, and complements upp er b ounds prov en recently [3, 21]. The pro of uses the magnetization as a test function giving a lo wer b ound on the sp ectral gap, using the fact that lo wer-bounding the v ariance of the magnetization is enough to show a low er b ound for the relaxation time (see e.g. [13, 18]). The details are giv en in subsection 3.3. Corollary 1.3. F or the c ontinuous time Glaub er dynamics wher e e ach vertex is up date d at r ate 1: 1. On the r andom d -r e gular gr aph gr aph with β = β c , ther e exists some c onstant c 0 > 0 such that with pr ob ability 1 − o (1) , the mixing time of the Glaub er dynamics satisﬁes t mix ≥ c 0 √ n . 2. On the r andom Er dös-R ényi r andom gr aph G ( n, d/n ) , at β = β c , for every ε > 0 ther e exists δ > 0 such that P [ t mix ≥ δ √ n ] ≥ 1 − ε . Recen tly , Co ja-Oghlan et. al. [5] studied the ﬂuctuations of the free energy of the Ising mo del in the high and low-temperature regimes and ask ed whether it is p ossible to study the ﬂuctuations at criticalit y . Our next corollary answ ers this question: Corollary 1.4. Consider the Ising mo del on G ( n, d/n ) at the critic al temp er atur e and let Z n := X σ ∈{− 1 , 1 } n exp β X u ∼ v σ u σ v ! 5 b e its p artition function. If we set ∆ n := log( Z n ) − n log(2) − | E n | log cosh( β ) − log 4 √ n √ 2 π + 3 4 , wher e | E n | is the numb er of e dges of the gr aph, then ∆ n ( d ) − − − → n →∞ ∞ X i =3  ˜ C i log(1 + d − i ) − d i + 1 2 i  + log Z R exp y 2 p 2( d − 1) X −  1 12 + 1 4( d − 1)  y 4 ! d y wher e the ˜ C i ∼ Pois ( d i 2 i ) ar e indep endent, and also indep endent of X ∼ N (0 , 1) . Remark 1.5. L et Y b e the limit describ e d ab ove. It is also true that   ∆ n , | E n | − 1 2 dn q 1 2 dn   ( d ) − − − → n →∞ ( Y , X 1 ) , wher e ( X, X 1 ) is a Gaussian ve ctor, indep endent of the ˜ C i , with X , X 1 ∼ N (0 , 1) and Cov ( X , X 1 ) = q d − 1 d . Inde e d in this c ase, X 1 is the limit of the normalize d e dge c ount and X is the limit of the normalize d of the long p ath c ount, and one c ould pr ove that ( X , X 1 ) has the ab ove pr op erties, with a c alculation similar to that p erforme d in the pr o of of Pr op osition 3.8. However, to ke ep the p ap er to a manage able length, we do not pursue that her e. A c kno wledgmen ts W e would like to thank Nic k W ormald and Eyal Lubetzky for helpful discussions. The work w as partially sup p orted by a Simons In v estigator gran t. 2 Preliminaries In the en tirety of this pap er, whenever w e write a n ∼ b n for tw o sequences a n , b n , we mean that a n /b n → 1 as n → ∞ . Also, A ≍ B will mean that c 1 B ≤ A ≤ c 2 B for some absolute constan ts c 1 , c 2 > 0 . 2.1 Deﬁnitions W e start b y in tro ducing the Ising mo del and the Glaub er dynamics. Deﬁnition 2.1. • L et G b e a gr aph with vertex set V . The Ising mo del on this gr aph with inverse temp er atur e β is the pr ob ability me asur e µ on {− 1 , 1 } V satisfying µ β ( σ ) = 1 Z β ,G exp β X u ∼ v σ u σ v ! . 6 • Fix a me asur e ν on {− 1 , 1 } V . F or a c onﬁgur ation σ ∈ {− 1 , 1 } V and for some v ∈ V , let σ ⊕ v b e the c onﬁgur ation σ in which the spin at vertex v is ﬂipp e d. F or two c onﬁgur ations σ, τ ∈ {− 1 , 1 } V , we write σ ∼ τ if τ = σ ⊕ v for some v ∈ V . The c ontinuous time Glaub er Dynamics on G for ν at r ate 1 is a Markov chain with tr ansition r ates, for σ  = τ , e qual to q ( σ, τ ) =          ν ( τ ) ν ( σ ) + ν ( τ ) , if σ ∼ τ 0 , otherwise . Informal ly, every vertex gets up date d at r ate one, and the up date happ ens c onditional ly on the c onﬁgur ation in the r est of the vertic es. W e contin ue with the p -W asserstein distance. The sp ecial p = 1 case is the metric used in Theorems 1.1, 1.2. F or a metric space ( X , d ) , w e denote b y M 1 ( X ) to be the set of probabilit y measures on X . Deﬁnition 2.2. The p -W asserstein distanc e asso ciate d to d is the function W p ( µ, ν ) := inf γ ∈C ( µ,ν )  E ( x,y ) ∼ γ d ( x, y ) p  1 /p , wher e the inf is taken over al l c ouplings γ of µ and ν . 2.2 Momen ts of partition functions in regular graphs Let G b e a d -regular graph on n v ertices. W e will call a measure µ on X V a homogeneous spin system if it is of the form µ ( σ ) ∝ Y u ∈ V ¯ ψ ( σ u ) Y u ∼ v ψ ( σ u , σ v ) , where ψ : X 2 → [0 , ∞ ) and ¯ ψ : X → [0 , ∞ ) are functions, called the weigh ts of the spin system. Supp ose we are giv en a ﬁnite spin set X and weigh ts ( ψ , ψ ) with ψ > 0 . F or a conﬁguration σ ∈ X V , its edge-empirical distribution is h σ ( x, x ′ ) := 1 dn X u ∈ V X v ∈ ∂ u 1 σ u = x, σ v = x ′ . Supp ose G is a d -regular graph. Then, the edge-empirical distribution uniquely determines the vertex-empirical distribution as well: h σ ( x ) := 1 n X u ∈ V 1 σ u = x = X x ′ ∈X h σ ( x, x ′ ) . Let H n b e the set of h ∈ [0 , 1] X 2 that are p ossible edge-empirical distributions (for some graph G ) and H n b e the set of h ∈ [0 , 1] X that are p ossible v ertex-empirical distributions. 7 Also, for an y h ∈ H n let A h b e the set of conﬁgurations that ha ve edge-empirical density h . Then, under the d -regular conﬁguration mo del, E | A h | = 1 ( dn − 1)!! · n ! Q x ∈X ( nh ( x ))! · Y x ∈X ( dnh ( x ))! Q x ′ ∈X ( dnh ( x, x ′ ))! · Y x ∈X ( dnh ( x, x ) − 1)!! · Y x ′  = x p ( dnh ( x, x ′ ))! ! . (3) The pro of w e present here can b e found, for example, as an in termediate step for proving Theorem 3 in [7]. If w e ﬁx an edge-empirical distribution, there are n ! Q x ∈X ( nh ( x ))! w ays to c ho ose the spins of the v ertices. Then, there are Q x ∈X ( dnh ( x ))! Q x ′ ∈X ( dnh ( x,x ′ ))! w ays to choose the spin at the half-edges and then another Q x ∈X ( dnh ( x, x ) − 1)!! · Q x ′  = x p ( dnh ( x, x ′ ))! ! admissible wa ys to ﬁnd a matching of the half-edges. W e will use the fact that if Z h = X σ ∈ A h Y u ∈ V ψ ( σ u ) Y { u,v }∈ E ψ ( σ u , σ v ) is the con tribution of conﬁgurations σ with h ( σ ) = h to the partition function, then for ev ery h ∈ H n , E ( Z h ) = exp  n  ⟨ h, log ψ ⟩ + d 2 ⟨ h, log ψ ⟩  · E | A h | . (4) It is imp ortant to note that due to Stirling’s form ula, due to (3) for any h , E ( Z h ) ≍ n Θ(1) · exp( n Φ( h )) , where Φ( h ) = d 2 H ( h ) − ( d − 1) H ( ¯ h ) + d 2 ⟨ h, log ψ ⟩ + ⟨ ¯ h, log ¯ ψ ⟩ . (5) Here, w e denote by H ( p ) the Shannon entrop y of a probability vector p . Therefore, the h ∈ H n that are close to the maximizer of Φ are naturally the ones whose contributions dominate the exp ectation of the partition function. This in tuition will b e formalized and used rep eatedly in sections 4 and 5. 3 Pro ofs of main Theorems 3.1 Pro of of Theorem 1.1 In this subsection, we pro ve Theorem 1.1. W e will w ork with the d -regular conﬁguration mo del on n vertices instead. Since the probabilit y of simplicit y of the conﬁguration mo del con verges to a p ositiv e constant, pro ving Theorem 1.1 for the conﬁguration mo del implies 8 that it holds for G n,d as well. A t ﬁrst, we state a useful lemma which will later imply that n 3 / 4 is the righ t scaling for the magnetization. Its pro of will b e in Section 4. Lemma 3.1. F or every ε > 0 , ther e exists C = C ( ε ) > 0 , such that P   X σ : | m ( σ ) | >C n 3 / 4 Z σ ≥ 1 2 r d − 2 d − 1 ε 2 · E   X σ : | m ( σ ) |≤ C n 3 / 4 Z σ     ≤ ε. T o pro v e Theorem 1.1, we will need a relation betw een the second momen t and the ﬁrst momen t of Z m and the distribution of cycle coun ts under the regular and the planted measure. Lemma 3.2. L et C > 0 and m = m ( n ) b e a se quenc e of p ositive inte gers with the same p arity as n , with | m | ≤ C n 3 / 4 . Then, E ( Z m ) ∼ c d,n · exp  − ( d − 1)( d − 2) 12 d 2 · m 4 n 3  , wher e c d,n = 2 1 / 2 √ d − 1 √ n √ π d · (2 · cosh( β ) d/ 2 ) n , and E ( Z 2 m ) E ( Z m ) 2 − − − → n →∞ r d − 1 d − 2 . Mor e over, b oth of these c onver genc es ar e uniform in m . F or eac h i ≥ 3 , let Y i,n b e the num b er of cycles of length i presen t in G . F or i = 2 , Y 2 ,n is the num b er of double edges in G , whereas Y 1 ,n is the num b er of lo ops in G . Regarding the distribution of Y i,n , we prov e the follo wing prop osition: Prop osition 3.3. 1. F or e ach i ≥ 1 , Y i,n ( d ) − − − → n →∞ C i ∼ Pois  ( d − 1) i 2 i  . These c onver genc es hold jointly, and the limits C i ar e indep endent. 2. L et m b e as in L emma 3.2. Under the plante d me asur e P ∗ m , for any i ≥ 1 , Y i,n ( d ) − − − → n →∞ C ′ i ∼ Pois  ( d − 1) i + 1 2 i  . As b efor e, these c onver genc es hold jointly, and the limits ar e indep endent. 9 In view of Prop osition 3.3, we set δ i = ( d − 1) − i , λ i = ( d − 1) i 2 i and for any k ∈ N , W k,n = k X i =1 ( Y i,n log(1 + δ i ) − λ i δ i ) , W ( R ) k,n = sgn ( W k,n ) min( | W k,n | , R ) W k = k X i =1 ( C i log(1 + δ i ) − λ i δ i ) , W ( R ) k = sgn ( W k ) min( | W k | , R ) , W ′ k = k X i =1 ( C ′ i log(1 + δ i ) − λ i δ i ) , W ′ ( R ) k = sgn ( W ′ k ) min( | W ′ k | , R ) and W ∞ = ∞ X i =1 ( C i log(1 + δ i ) − λ i δ i ) , and W ′ ∞ = ∞ X i =1 ( C ′ i log(1 + δ i ) − λ i δ i ) . Remark 3.4. • F or any δ 0 ∈ (0 , 1) and k ∈ N , P  e W k ≤ δ 0  = P  e − W k ≥ δ − 1 0  ≤ δ 0 · E ( e − W k ) = δ 0 · exp k X i =1 λ i δ 2 i 1 + δ i ! ≤ δ 0 · r d − 1 d − 2 . (6) • Sinc e W ( R ) k ≥ W k when W k < 0 , the same ine quality holds for W ( R ) k as wel l. • exp(2 W k ) − − − → k →∞ exp(2 W ∞ ) and exp( W ′ k ) − − − → k →∞ exp( W ′ ∞ ) , b oth a.s. and in L 2 , as was note d in [14] . Our main claim is the following: Prop osition 3.5. F or any k ∈ N and C > 0 , lim sup n →∞ sup | m |≤ C n 3 / 4 E "  Z m E ( Z m ) − exp  W ( R ) k,n   2 # ≤ α k,R ,C wher e lim R →∞ lim k →∞ α k,R ,C = 0 for any C > 0 . W e explain wh y Prop osition 3.5 is enough to ﬁnish the pro of of Theorem 1.1. Pr o of of The or em 1.1. Let ε ∈ (0 , 1) . Choose C > 0 large enough, so that the statement of Lemma 3.1 holds and so that µ reg ([ − C, C ]) > 1 − ε . Also, let − C = y 0 < y 1 < · · · < y s = C b e real num b ers so that for an y i = 0 , 1 , . . . , s − 1 , µ reg ([ y i , y i +1 ]) < ε. By Sk orokho d’s represen tation Theorem, it is possible couple the relev ant random v ariables, so that the con vergences in Prop osition 3.3 are almost sure conv ergences. Note that the cou- pling implied in the W asserstein distance in this case is trivialized, since µ reg is deterministic. Cho ose δ 1 , δ 2 , δ > 0 such that: P (exp( W k ) ≤ δ 1 ) ≤ ε, 2 δ 2 δ 1 − δ 2 ≤ ε, and sδ δ 2 2 ≤ ε. (7) 10 Due to (6), it is p ossible to set δ 1 = q d − 2 d − 1 · ε . Since ε ∈ (0 , 1) , δ 2 < δ 1 / 2 . Also, for eac h i ∈ { 1 , 2 , . . . , s } , let Z ( i ) = X − C n 3 / 4 ≤ m ≤ y i n 3 / 4 Z m . Because of Prop osition 3.2, w e can set sup n sup | m |≤ C n 3 / 4 E ( Z m ) inf | m |≤ C n 3 / 4 E ( Z m ) =: M < ∞ . F or any i , set N i = # { m : − C n 3 / 4 ≤ m ≤ y i n 3 / 4 } . Due to Prop osition 3.5, for ev ery i , if n is large enough, E   Z ( i ) − exp  W ( R ) k,n  E ( Z ( i ) )  2  = E     X − C n 3 / 4 ≤ m ≤ y i n 3 / 4 ( Z m − exp  W ( R ) k,n  E ( Z m ))   2   ≤ N 2 i · sup − C n 3 / 4 ≤ m ≤ y i n 3 / 4 E   Z m − exp( W ( R ) k,n ) E ( Z m )  2  ≤ N 2 i · 2 α k,R ,C · sup − C n 3 / 4 ≤ m ≤ y i n 3 / 4 E ( Z m ) ! 2 ≤ 2 M 2 α k,R ,C  E ( Z ( i ) )  2 . Therefore, if k , R and n are large enough, E "  Z ( i ) E ( Z ( i ) ) − exp  W ( R ) k,n   2 # ≤ δ. (8) Consider the even t A (1) n := n exp  W ( R ) k,n  > δ 1 o ∩ s \ i =1      Z ( i ) E ( Z ( i ) ) − exp  W ( R ) k,n      ≤ δ 2  . On the even t A (1) n , Z ( s ) E ( Z ( s ) ) ≥ δ 1 − δ 2 ≥ 1 2 r d − 2 d − 1 ε. Therefore, on the ev ent A n = A (1) n ∩    X σ : | m ( σ ) | >C n 3 / 4 Z σ ≤ 1 2 r d − 2 d − 1 ε 2 · E  Z ( s )     , it is true that P σ : | m ( σ ) | >C n 3 / 4 Z σ Z < P σ : | m ( σ ) | >C n 3 / 4 Z σ Z ( s ) ≤ ε ⇒ Z − Z ( s ) Z < ε. 11 W e pro ve that for large n , P ( A n ) ≥ 1 − 4 ε . Due to (8) and Chebyshev’s inequalit y , P s [ i =1      Z ( i ) E ( Z ( i ) ) − exp  W ( R ) k,n      > δ 2  ! ≤ s · δ δ 2 2 ≤ ε. Also, by our c hoice of δ 1 , if n is large enough, P  exp  W ( R ) k,n  ≤ δ 1  ≤ 2 ε, whic h indeed implies that P ( A c n ) ≤ 4 ε . Our claim is that on the ev ent A n , d KS ( µ n, reg , µ reg ) < 10 ε. (9) This will imply the result, as E [ d KS ( µ n, reg , µ reg )] ≤ 10 ε + P ( A c n ) < 14 ε if n is large enough, since d KS ( µ, ν ) ≤ 1 for any measures µ, ν due to the deﬁnition of d KS . W e ﬁrst pro v e that on the even t A n , sup 1 ≤ i ≤ s | µ n, reg ([ − C, y i ]) − µ reg ([ − C, y i ]) | < 4 ε. (10) Indeed, | µ n, reg ([ − C, y i ]) − µ reg ([ − C, y i ]) | =       Z ( i ) Z − R [ − C,y i ] exp  − ( d − 1)( d − 2) 12 d 2 u 4  d u R R exp  − ( d − 1)( d − 2) 12 d 2 u 4  d u       ≤     Z ( i ) Z − Z ( i ) Z ( s )     +       R [ − C,y i ] exp  − ( d − 1)( d − 2) 12 d 2 u 4  d u R [ − C,C ] exp  − ( d − 1)( d − 2) 12 d 2 u 4  d u − R [ − C,y i ] exp  − ( d − 1)( d − 2) 12 d 2 u 4  d u R R exp  − ( d − 1)( d − 2) 12 d 2 u 4  d u       +     Z ( i ) Z ( s ) − E ( Z ( i ) ) E ( Z ( s ) )     +       E ( Z ( i ) ) E ( Z ( s ) ) − R [ − C,y i ] exp  − ( d − 1)( d − 2) 12 d 2 u 4  d u R [ − C,C ] exp  − ( d − 1)( d − 2) 12 d 2 u 4  d u       . W e sho w that on the even t A n , each one of these terms is < ε . 1. Since Z − Z ( s ) Z < ε , and due to the wa y C was c hosen, the ﬁrst tw o terms are < ε . 2. When n → ∞ , due to Lemma 3.2, the last term is deterministic and conv erges to 0: E ( Z i ) ∼ c d,n X − C n 3 / 4 ≤ m ≤ y i n 3 / 4 exp  − ( d − 1)( d − 2) 12 d 2 · m 4 n 3  ∼ c d,n n 3 / 4 2 Z y i − C exp  − ( d − 1)( d − 2) 12 d 2 u 4  d u, b ecause of the appro ximation of the integral by a Riemann sum. 12 3. On the even t A n , Z ( s ) ≥ ( δ 1 − δ 2 ) · E ( Z ( s ) ) and     Z ( i ) E ( Z ( i ) ) − Z ( s ) E ( Z ( s ) )     ≤     Z ( i ) E ( Z ( i ) ) − exp  W ( R ) k,n      +     Z ( s ) E ( Z ( s ) ) − exp  W ( R ) k,n      ≤ 2 δ 2 , whic h implies     Z ( i ) Z ( s ) − E ( Z ( i ) ) E ( Z ( s ) )     ≤     Z ( i ) E ( Z ( i ) ) − Z ( s ) E ( Z ( s ) )     · E ( Z ( s ) ) Z ( s ) ≤ 2 δ 2 δ 1 − δ 2 ≤ ε. W e hav e successfully pro ven (10). Let y ∈ R . If | y | > C , due to the w ay C was c hosen, if n is large enough, | µ n, reg (( −∞ , y ]) − µ reg (( −∞ , y ]) | < ε . On the other hand, if | y | ≤ C , | µ n, reg (( −∞ , y ]) − µ reg (( −∞ , y ]) | ≤ ε + | µ n, reg ([ − C, y ]) − µ reg ([ − C, y ]) | and there exists 0 ≤ i ≤ s − 1 suc h that y i ≤ y ≤ y i +1 . F or this i , w e claim that µ n, reg ([ − C, y ]) and µ reg ([ − C, y ]) lie in the interv al I = [ µ reg ([ − C, y i ]) − 4 ε, µ reg ([ − C, y i +1 ]) + 4 ε ] . This will mean that their diﬀerence is at most the length of the interv al, whic h is ≤ 9 ε , concluding the pro of. This fact is obvious for µ reg ([ − C, y ]) . As for µ n, reg ([ − C, y ]) , observe that on the ev ent A n , if n is large enough, µ reg ([ − C, y i ]) − 4 ε ≤ µ n, reg ([ − C, y i ]) ≤ µ n, reg ([ − C, y ]) ≤ µ n, reg ([ − C, y i +1 ]) ≤ µ reg ([ − C, y i +1 ]) + 4 ε. This completes the pro of of Theorem 1.1. It, therefore, remains to show Prop osition 3.5, giv en Lemma 3.2 and Prop osition 3.3. Pr o of of Pr op osition 3.5. W e show the Prop osition with α k,R ,C := r d − 1 d − 2 − 2 · E h exp  W ′ ( R ) k i + E h exp  2 W ( R ) k i . Due to the dominated conv ergence theorem and the in tegrability of exp( W ′ ∞ ) , exp(2 W ∞ ) , lim R →∞ lim k →∞ E h exp  W ′ ( R ) k i = ∞ Y i =1 e − λ i δ i E  (1 + δ i ) C ′ i  = ∞ Y i =1 exp  λ i δ 2 i  = r d − 1 d − 2 and lim R →∞ lim k →∞ E h exp  2 W ( R ) k i = ∞ Y i =1 e − 2 λ i δ i E  (1 + δ i ) 2 C i  = ∞ Y i =1 exp( λ i δ 2 i ) = r d − 1 d − 2 , so the condition claimed ab out the α k,R ,C holds. F or an y sequence of m = m ( n ) as in Lemma 3.2, E "  Z m E ( Z m ) − exp  W ( R ) k,n   2 # = E ( Z 2 m ) E ( Z m ) 2 − 2 · E ∗ m  exp  W ( R ) k,n  + E  exp  2 W ( R ) k,n  → n →∞ α k,R ,C whic h implies the result. 13 3.2 Pro of of Theorem 1.2 In this subsection, w e complete the pro of of Theorem 1.2. In this case, w e study a normalized v ersion of the partition function. This is necessary; without it, pro ving concentration is imp ossible, due to, for example, the ﬂuctuations in the n umber of edges. F or σ ∈ {− 1 , 1 } n , let ˜ Z σ = 2 − n Y 1 ≤ u 0 lim inf n →∞ P | m |≤ n 3 / 4 E ( ˜ Z m ) P | m |≤ C n 3 / 4 E ( ˜ Z m ) . Then, ˜ c 0 > 0 and for any ε > 0 , ther e exists C = C ( ε ) > 0 such that P   X | m | >C n 3 / 4 ˜ Z m ≥ ˜ c 0 4 r d d − 1 ε 2 exp − ε − 1 p 2( d − 1) − 1 4( d − 1) ! · E   X | m |≤ C n 3 / 4 ˜ Z m     < ε. As in the previous subsection, we compute the ﬁrst and second moments of ˜ Z m . Lemma 3.7. L et m = m ( n ) b e a se quenc e of p ositive inte gers with the same p arity as n , with | m | ≤ C n 3 / 4 . Then, E ( ˜ Z m ) ∼ √ 2 √ π n · exp  − 1 12 · m 4 n 3 − 3 4  and E ( ˜ Z 2 m ) E ( ˜ Z m ) 2 ∼ exp 1 2( d − 1) · m 4 n 3 + ∞ X i =3 d − i 2 i ! . Mor e over, these c onver genc es ar e uniform in m . Let X ℓ,n denote the num b er of paths of length ℓ p resen t in G and b X ℓ,n := X ℓ,n − 1 2 nd ℓ q 1 2 nd 2 ℓ ℓ 2 d − 1 , 14 as introduced in (2). Moreo ver, for an y R > 0 , set b X ( R ) ℓ,n = sgn ( b X ℓ,n ) min( | b X ( R ) ℓ,n | , R ) . F or eac h i ≥ 3 , let ˜ Y i,n b e the n umber of cycles of length i presen t in G . The second ingredien t we will need is the asymptotic distribution of b X ℓ,n and ˜ Y i,n b oth under the regular and the planted measure. Prop osition 3.8. 1. F or e ach i ≥ 3 , ˜ Y i,n ( d ) − − − → n →∞ ˜ C i ∼ Pois  d i 2 i  . These c onver genc es hold jointly, and the limits C i ar e indep endent. 2. F or any ℓ ∈ N , b X ℓ,n ( d ) − − − → n →∞ X ℓ ∼ N  0 , 1 + γ (2) ℓ  , wher e γ (2) ℓ := d − 1 ℓ 2 ℓ X k =1  ( ℓ − k + 1) 2 · d − k  − 1 − − − → ℓ →∞ 0 . A lso, this c onver genc e holds jointly with the c onver genc es in 1 and the limits ar e inde- p endent. 3. L et m b e as in L emma 3.7. Under the plante d me asur e P ∗ m , for any i ≥ 3 ˜ Y i,n ( d ) − − − → n →∞ ˜ C ′ i ∼ Pois  d i + 1 2 i  . As b efor e, these c onver genc es hold jointly, and the limits ar e indep endent. 4. L et m b e as in L emma 3.7, mor e over satisfying the c ondition that m · n − 3 / 4 − − − → n →∞ x . Under the plante d me asur e P ∗ m , for any ℓ ∈ N b X ℓ,n ( d ) − − − → n →∞ X ′ ℓ ∼ N x 2 p 2( d − 1)  1 + γ (1) ℓ  , 1 + γ (2) ℓ ! , wher e γ (1) ℓ := d − 1 ℓ ℓ X k =1  ( ℓ − k + 1) · d − k  − 1 − − − → ℓ →∞ 0 . A lso, this c onver genc e holds jointly with the c onver genc es in 3 and the limits ar e inde- p endent. 15 In view of 3, w e set ˜ δ i = d − i , ˜ λ i = d i 2 i and ˜ µ i = ˜ λ i (1 + ˜ δ i ) . Also, just as in the pro of of Theorem 1.1, let ˜ W k,n := k X i =3  ˜ Y i,n log(1 + ˜ δ i ) − ˜ λ i ˜ δ i  , ˜ W ( R ) k,n = sgn ( ˜ W k,n ) min( | ˜ W k,n | , R ) ˜ W k = k X i =3  ˜ C i log(1 + ˜ δ i ) − ˜ λ i ˜ δ i  , ˜ W ( R ) k = sgn ( ˜ W k ) min( | ˜ W k | , R ) and ˜ W ′ k = k X i =3  ˜ C ′ i log(1 + ˜ δ i ) − ˜ λ i ˜ δ i  and ˜ W ′ ( R ) k = sgn ( ˜ W ′ k ) min( | ˜ W ′ k | , R ) . Our main claim is the following: Prop osition 3.9. F or any ℓ, k ∈ N and R, C > 0 , lim sup n →∞ sup | m |≤ C n 3 / 4 E   ˜ Z m E ( ˜ Z m ) − exp  θ m b X ( R ) ℓ,n − θ 2 m / 2 + ˜ W ( R ) k,n  ! 2   ≤ ˜ α ℓ,k,R ,C wher e θ m = ( m · n − 3 / 4 ) 2 p 2( d − 1) and lim R →∞ lim ℓ →∞ lim k →∞ ˜ α ℓ,k,R ,C = 0 for any C > 0 . W e explain ho w this Prop osition implies Theorem 1.2. Pr o of of The or em 1.2. Due to the Sk orokho d’s represen tation Theorem, we may couple all the graphs and random v ariables such that the con vergences in Prop osition 3.8 are almost sure conv ergences. W e claim that if ℓ, R and n are large enough, then d KS  µ n, ER , µ ( X ( R ) ℓ )  ≤ 18 ε. This will imply the desired result: If ℓ and R are large enough, we can couple X ( R ) ℓ with a X ∼ N (0 , 1) random v ariable such that X ( R ) ℓ = X with probabilit y ≥ 1 − ε . So, W 1  µ ( X ( R ) ℓ ) , µ  ≤ P ( X ( R ) ℓ  = X ) ≤ ε, whic h means that if n is large enough, W 1 ( µ n, ER , µ ) ≤ 19 ε, concluding the pro of of Theorem 1.2. W e, therefore, hav e to pro ve this claim. 16 Cho ose ℓ ∈ N and C > 0 large enough, so that Lemma 3.6 holds and both {| X ℓ | ≥ ε − 1 } and { µ ( X ( R ) ℓ ) ([ − C, C ]) ≤ 1 − ε } hav e probabilit y ≤ ε . Observe that if | x | ≤ ε − 1 , for any y ∈ R , x p 2( d − 1) · y 2 −  1 4( d − 1) + 1 12  · y 4 ≤ x 2 2 ≤ ε − 2 2 and Z R exp x p 2( d − 1) · y 2 −  1 4( d − 1) + 1 12  · y 4 ! d y ≥ e − 1 4( d − 1) − 1 12 Z 1 0 exp x p 2( d − 1) · y 2 ! d y ≥ exp − 1 4( d − 1) − 1 12 − ε − 1 p 2( d − 1) ! . This means that there exists a constan t D = D ( ε ) > 0 suc h that for any y 1 < y 2 , R y 2 y 1 exp x p 2( d − 1) · u 2 −  1 4( d − 1) + 1 12  · u 4 ! d u R R exp x p 2( d − 1) · u 2 −  1 4( d − 1) + 1 12  · u 4 ! d u ≤ D ( y 2 − y 1 ) , therefore there exist − C = y 0 < y 1 < · · · < y s = C suc h that for any i ∈ { 0 , 1 , . . . , s − 1 } , P  µ ( X ( R ) ℓ ) ([ y i , y i +1 ]) > ε | X ( R ) ℓ ∈ [ − ε − 1 , ε − 1 ]  = 0 . Cho ose η 1 = q d − 1 d · ε, η 2 , η > 0 such that: P (exp( ˜ W k ) ≤ η 1 ) ≤ ε, η 2 ≤ ˜ c 0 4 r d − 1 d ε 2 exp − ε − 1 p 2( d − 1) − 1 4( d − 1) ! , 2 η 2 η 1 ˜ c 1 − η 2 ≤ ε and sη η 2 2 ≤ ε, where ˜ c 1 := exp − C 2 ε − 1 p 2( d − 1) − C 4 4( d − 1) ! > 0 . This choice is p ossible due to an argument almost identical to that of (6). Also, for each i ∈ { 1 , 2 , . . . , s } , let ˜ Z ( i ) = X − C n 3 / 4 ≤ m ≤ y i n 3 / 4 ˜ Z m . Because of Prop osition 3.7, w e can set sup n sup | m |≤ C n 3 / 4 E ( ˜ Z m ) inf | m |≤ C n 3 / 4 E ( ˜ Z m ) =: ˜ M < ∞ . 17 F or an y i , let ˜ N i = # { m : − C n 3 / 4 ≤ m ≤ y i n 3 / 4 } . Then, E     ˜ Z ( i ) − exp  ˜ W ( R ) k,n  X − C n 3 / 4 ≤ m ≤ y i n 3 / 4 E ( ˜ Z m ) · exp  θ m b X ( R ) ℓ,n − θ 2 m / 2    2   = E     X − C n 3 / 4 ≤ m ≤ y i n 3 / 4 h ˜ Z m − exp  W ( R ) k,n + θ m b X ( R ) ℓ,n − θ 2 m / 2  E ( ˜ Z m ) i   2   ≤ ˜ N 2 i · 2 ˜ α ℓ,k,R ,C · sup − C n 3 / 4 ≤ m ≤ y i n 3 / 4 E ( ˜ Z m ) ! 2 ≤ 2 ˜ M 2 ˜ α ℓ,k,R ,C · ( E ( ˜ Z ( i ) )) 2 . Therefore, for any 1 ≤ i ≤ s , if k , ℓ, R and n are large enough, E     ˜ Z ( i ) − exp  ˜ W ( R ) k,n  X − C n 3 / 4 ≤ m ≤ y i n 3 / 4 E ( ˜ Z m ) · exp  θ m b X ( R ) ℓ,n − θ 2 m / 2    2   ≤ η · E ( ˜ Z ( i ) ) 2 . (11) Consider the even t B (1) n := n | b X ( R ) ℓ,n | ≤ ε − 1 o ∩ { µ ( X ( R ) ℓ ) ([ − C, C ]) > 1 − ε } ∩ n exp  ˜ W ( R ) k,n  > η 1 o ∩ s \ i =1                 ˜ Z ( i ) − exp  ˜ W ( R ) k,n  P − C n 3 / 4 ≤ m ≤ y i n 3 / 4 E ( ˜ Z m ) exp( θ m b X ( R ) ℓ,n − θ 2 m / 2)      E ( ˜ Z ( i ) ) ≤ η 2            ∩        sup a,b ∈ [ − C,C ]         P an 3 / 4 ≤ m ≤ bn 3 / 4 E ( ˜ Z m ) exp( θ m b X ( R ) ℓ,n − θ 2 m / 2) ˜ c n R b a exp  y 2 √ 2( d − 1) X ( R ) ℓ −  1 12 + 1 4( d − 1)  y 4  d y − 1         ≤ ε 2        , where ˜ c n = 4 √ n √ 2 π · exp( − 3 4 ) . On the even t B (1) n , ˜ Z ( s ) ≥ exp  ˜ W ( R ) k,n  X | m |≤ C n 3 / 4 h E ( ˜ Z m ) exp( θ m b X ( R ) ℓ,n − θ 2 m / 2) i − η 2 · E ( ˜ Z ( s ) ) ≥ η 1 exp − ε − 1 p 2( d − 1) − 1 4( d − 1) ! X | m |≤ n 3 / 4 E ( ˜ Z m ) − η 2 · E ( ˜ Z ( s ) ) ≥ " η 1 exp − ε − 1 p 2( d − 1) − 1 4( d − 1) ! ˜ c 0 2 − η 2 # · E ( ˜ Z ( s ) ) ≥ ˜ c 0 4 r d − 1 d · ε exp − ε − 1 p 2( d − 1) − 1 4( d − 1) ! · E ( ˜ Z ( s ) ) , 18 due to the w ay ˜ c 0 , η 1 and η 2 w ere deﬁned. Therefore, on the even t B n := B (1) n ∩    X | m | >C n 3 / 4 ˜ Z m ≤ ˜ c 0 4 r d d − 1 ε 2 exp − ε − 1 p 2( d − 1) − 1 4( d − 1) ! · E ( ˜ Z ( s ) )    w e will h a ve that ˜ Z − ˜ Z ( s ) ˜ Z < ε . W e observe that on the ev ent B (1) n , since − C ≤ y i ≤ C ,         P − C n 3 / 4 ≤ m ≤ y i n 3 / 4 E ( ˜ Z m ) exp( θ m b X ( R ) ℓ,n − θ 2 m / 2) P | m |≤ C n 3 / 4 E ( ˜ Z m ) exp( θ m b X ( R ) ℓ,n − θ 2 m / 2) − µ ( X ( R ) ℓ ) ([ − C, y i ]) µ ( X ( R ) ℓ ) ([ − C, C ])         ≤ ε. W e claim that for large n , P ( B n ) ≥ 1 − 8 ε . Due to (11) and Chebyshev’s inequalit y , P       s [ i =1                 ˜ Z ( i ) − exp  ˜ W ( R ) k,n  P − C n 3 / 4 ≤ m ≤ y i n 3 / 4 E ( ˜ Z m ) exp( θ m b X ( R ) ℓ,n − θ 2 m / 2)      E ( ˜ Z ( i ) ) > η 2                  ≤ s · η η 2 2 ≤ ε. Also, due to the wa y η 1 w as c hosen, if n is large enough, P  exp  ˜ W ( R ) k,n  ≤ η 1  ≤ 2 ε and P  | b X ( R ) ℓ,n | > ε − 1  ≤ 2 ε. Moreo ver, for any sequence ( x n ) ∞ n =1 with | x n | ≤ ε − 1 whic h conv erges to x , uniformly for a, b ∈ [ − C , C ] it holds that X an 3 / 4 ≤ m ≤ bn 3 / 4 E ( ˜ Z m ) exp( θ m x n − θ 2 m / 2) ∼ ˜ c n Z b a exp  y 2 √ 2( d − 1) x − ( 1 4( d − 1) + 1 12 ) y 4  d y . Applying this for the sequence b X ( R ) ℓ,n − − − → n →∞ X ( R ) ℓ w e get that P ( B (1) n ) ≥ 1 − 7 ε for large enough n and therefore Lemma 3.6 implies that P ( B n ) ≥ 1 − 8 ε . W e will pro v e that on the even t B n , sup 1 ≤ i ≤ s | µ n, ER ([ − C, y i ]) − µ ( X ( R ) ℓ ) ([ − C, y i ]) | < 4 ε. (12) With an argument identical to that included in the pro of of Theorem 1.1, one can see that 19 this is enough to ﬁnish the pro of. Indeed, | µ n, ER ([ − C, y i ]) − µ ( X ( R ) ℓ ) ([ − C, y i ]) | =      ˜ Z ( i ) ˜ Z − µ ( X ( R ) ℓ ) ([ − C, y i ])      ≤      ˜ Z ( i ) ˜ Z − ˜ Z ( i ) ˜ Z ( s )      +      µ ( X ( R ) ℓ ) ([ − C, y i ]) − µ ( X ( R ) ℓ ) ([ − C, y i ]) µ ( X ( R ) ℓ ) ([ − C, C ])      +         ˜ Z ( i ) ˜ Z ( s ) − P − C n 3 / 4 ≤ m ≤ y i n 3 / 4 E ( ˜ Z m ) exp( θ m b X ( R ) ℓ,n − θ 2 m / 2) P − C n 3 / 4 ≤ m ≤ C n 3 / 4 E ( ˜ Z m ) exp( θ m b X ( R ) ℓ,n − θ 2 m / 2)         +         P − C n 3 / 4 ≤ m ≤ y i n 3 / 4 E ( ˜ Z m ) exp( θ m b X ( R ) ℓ,n − θ 2 m / 2) P − C n 3 / 4 ≤ m ≤ C n 3 / 4 E ( ˜ Z m ) exp( θ m b X ( R ) ℓ,n − θ 2 m / 2) − µ ( X ( R ) ℓ ) ([ − C, y i ]) µ ( X ( R ) ℓ ) ([ − C, C ])         . W e sho w that on the even t B n , each one of these terms is < ε . 1. On B n , ˜ Z − ˜ Z ( s ) ˜ Z < ε , therefore the ﬁrst t wo terms are < ε . 2. As we already explained, on B n , the last term is also < ε . 3. On the even t B n , ˜ Z ( s ) ≥ ( η 1 c 1 − η 2 ) · E ( ˜ Z ( s ) ) and         ˜ Z ( i ) ˜ Z ( s ) − P − C n 3 / 4 ≤ m ≤ y i n 3 / 4 E ( ˜ Z m ) exp( θ m b X ( R ) ℓ,n − θ 2 m / 2) P − C n 3 / 4 ≤ m ≤ C n 3 / 4 E ( ˜ Z m ) exp( θ m b X ( R ) ℓ,n − θ 2 m / 2)         ≤ 2 η 2 η 1 c 1 − η 2 ≤ ε. The pro of is complete. It remains to pro ve Prop osition 3.9. Pr o of of Pr op osition 3.9. W e prov e the desired statement, with ˜ α ℓ,k,R ,C := sup | x |≤ C ( exp x 4 2( d − 1) + ∞ X i =3 d − i 2 i ! − 2 E " exp x 2 p 2( d − 1) X ′ ( R ) ℓ − x 4 4( d − 1) + ˜ W ′ ( R ) k !# + E " exp √ 2 · x 2 √ d − 1 X ( R ) ℓ − x 4 2( d − 1) + 2 ˜ W ( R ) k !# ) . 20 This ˜ α ℓ,k,R ,C satisﬁes the desired prop ert y . Indeed, for any x ∈ [ − C , C ] ,      exp x 4 2( d − 1) + ∞ X i =3 d − i 2 i ! − E " exp x 2 p 2( d − 1) X ′ ( R ) ℓ − x 4 4( d − 1) + ˜ W ′ ( R ) k !#      ≤ exp  x 4 2( d − 1)       exp ∞ X i =3 d − i 2 i ! − E h exp  ˜ W ′ ( R ) k i      + E h exp  ˜ W ′ ( R ) k i      exp  x 4 2( d − 1)  − E " exp x 2 p 2( d − 1) X ′ ( R ) ℓ − x 4 4( d − 1) !#      . As explained in the pro of of Prop osition 3.5, lim R →∞ lim k →∞ E h exp  ˜ W ′ ( R ) k i = exp ∞ X i =3 d − i 2 i ! . Also, since      exp  x 4 2( d − 1)  − E " exp x 2 p 2( d − 1) X ′ ( R ) ℓ − x 4 4( d − 1) !#      ≤ exp  x 4 2( d − 1)      exp  x 4 4( d − 1) (2 γ (1) ℓ + γ (2) ℓ )  − 1     + E      exp x 2 p 2( d − 1) X ′ ℓ ! − exp x 2 p 2( d − 1) X ′ ( R ) ℓ !      ≤ exp  C 4 2( d − 1)  · C 4 d − 1 · (2 γ (1) ℓ + γ (2) ℓ ) + E " 1 | X ′ ℓ | >R exp x 2 p 2( d − 1) X ′ ℓ ! + 1 !# and E " 1 | X ′ ℓ | >R exp x 2 p 2( d − 1) X ′ ℓ ! + 1 !# ≤ v u u t 2 P ( | X ′ ℓ | > R ) · E " exp √ 2 · x 2 √ d − 1 X ′ ℓ ! + 1 # ≤ s 4 · P ( | X ′ ℓ | > R ) · exp  4 C 4 d − 1  , whic h hold if ℓ is large enough, it follo ws that lim R →∞ lim ℓ →∞ lim k →∞ sup | x |≤ C      exp x 4 2( d − 1) + ∞ X i =3 d − i 2 i ! − E " exp x 2 p 2( d − 1) X ′ ( R ) ℓ − x 4 4( d − 1) !# E h exp  ˜ W ′ ( R ) k i      = 0 . 21 In a similar fashion, we can pro ve that lim R →∞ lim ℓ →∞ lim k →∞ sup | x |≤ C      exp x 4 2( d − 1) + ∞ X i =3 d − i 2 i ! − E " exp √ 2 · x 2 √ d − 1 X ( R ) ℓ − x 4 2( d − 1) + 2 ˜ W ( R ) k !#      = 0 . This implies that indeed, lim R →∞ lim ℓ →∞ lim k →∞ ˜ α ℓ,k,R ,C = 0 . Supp ose Prop osition 3.9 is false. Then, there exists some sequence m = m ( n ) such that | m | ≤ C n 3 / 4 for any n and lim sup n →∞ E   ˜ Z m E ( ˜ Z m ) − exp  θ m b X ( R ) ℓ,n − θ 2 m / 2 + ˜ W ( R ) k,n  ! 2   > ˜ α ℓ,k,R ,C . Without loss of generalit y , we ma y assume (b y extracting a subsequence and, p ossibly , mo difying some terms) that m · n − 3 / 4 − − − → n →∞ x ∈ [ − C, C ] . Then, due to Lemma 3.7 and Prop osition 3.8, E   ˜ Z m E ( ˜ Z m ) − exp  θ m b X ( R ) ℓ,n − θ 2 m / 2 + ˜ W ( R ) k,n  ! 2   = E ( ˜ Z 2 m ) E ( Z m ) 2 − 2 · E ∗ m h exp  θ m b X ( R ) ℓ,n − θ 2 m / 2 + ˜ W ( R ) k,n i + E h exp  2 θ m b X ( R ) ℓ,n − θ 2 m + 2 ˜ W ( R ) k,n i → n →∞ exp x 4 2( d − 1) + ∞ X i =3 d − i 2 i ! − 2 E " exp x 2 p 2( d − 1) X ′ ( R ) ℓ − x 4 4( d − 1) + ˜ W ′ ( R ) k !# + E " exp √ 2 · x 2 √ d − 1 X ( R ) ℓ − x 4 2( d − 1) + 2 ˜ W ( R ) k !# . Due to the deﬁnition of ˜ α ℓ,k,R ,C that we gav e ab ov e, w e hav e reached a con tradiction and the pro of is complete. 3.3 Pro ofs of Corollaries In this subsection, w e pro ve Corollaries 1.3 and 1.4. Pr o of of Cor ol lary 1.3. As is w ell-known, the sp ectral gap of a Marko v chain ( X t ) t ≥ 0 , de- noted by gap ( X t ) , is deﬁned to b e gap ( X t ) := inf f  = c E ( f , f ) V ar π ( f ) , 22 where the inf is ov er all non-constant functions f : X → R and E ( f , f ) is the Dirichlet form of f , i.e. E ( f , f ) := 1 2 X x,y ∈X π ( x ) q ( x, y )( f ( x ) − f ( y )) 2 . W e set f ( σ ) = m ( σ ) = P v σ v . Then, E ( m, m ) ≤ 2 · X σ,τ π ( σ ) q ( σ, τ ) ≤ 2 n. In the regular graph case, due to Theorem 1.1, µ n, reg ( R \ ( − 1 , 1)) P − − − → n →∞ µ reg ( R \ ( − 1 , 1)) > 0 . If F n = { G : P ( | m | ≥ n 3 / 4 | G ) > µ reg ( R \ ( − 1 , 1)) / 2 } , then P G ∼ G n,d ( F c n ) − − − → n →∞ 0 and on the ev ent F n , V ar ( m ) ≥ n 3 / 2 · P ( | m | ≥ n 3 / 4 ) ≥ 1 2 µ reg ( R \ ( − 1 , 1)) · n 3 / 2 . This implies that gap ( X t ) ≤ E ( m, m ) V ar ( m ) ≤ cn − 1 / 2 . Due to the well-kno wn connection b etw een the sp ectral gap and the mixing time, it follo ws that t mix (1 / 4) ≥ c 1 · gap − 1 ( X t ) ≥ c 0 √ n and the ﬁrst statemen t follo ws. As for the Erdös-Rén yi case, let ε > 0 . Let K > 0 b e such that P ( | X | ≤ K ) ≥ 1 − ε 2 . Also, for some constan t c ( K ) , µ ( x ) ( R \ ( − 1 , 1)) ≥ c ( K ) for an y | x | ≤ K . Finally , if n is large enough, there exists a coupling of µ n and µ such that E ( d KS ( µ n , µ )) ≤ εc ( K ) 8 . Then, P ( d KS ( µ n , µ ) > c ( K ) / 4) ≤ ε/ 2 ⇒ P [ | µ n ( R \ ( − 1 , 1)) − µ ( R \ ( − 1 , 1)) | > c ( K ) / 2] ≤ ε/ 2 . Therefore, with probabilit y ≥ 1 − ε , w e kno w that µ n ( R \ ( − 1 , 1)) ≥ c ( K ) / 2 , and, with an argumen t similar to the one for the regular case, on this ev en t it is true that V ar ( m ) ≥ δ 1 n 3 / 2 , whic h then implies the second statemen t of the Corollary . W e mo ve on to Corollary 1.4. Pr o of of Cor ol lary 1.4. W e revisit the pro of of Theorem 1.2. Let E n b e the num b er of edges of the graph. Supp ose all the relev ant random v ariables are in the same probabilit y space. 23 Observ e that Z n = 2 n cosh( β ) | E n | · ˜ Z and if w e set ∆ ′ n := ∆ n − ∞ X i =3  ˜ C i log(1 + d − i ) − d i + 1 2 i  − log Z R exp y 2 p 2( d − 1) X −  1 12 + 1 4( d − 1)  y 4 ! d y , then {| ∆ ′ n | ≥ 6 ε } ⊆ {| log ˜ Z − log ˜ Z ( s ) | ≥ ε } ∪          log ˜ Z ( s ) − ˜ W ( R ) k,n − log X | m |≤ C n 3 / 4 E ( ˜ Z m ) exp( θ m b X ( R ) ℓ,n − θ 2 m / 2)       ≥ ε    ∪ n | ˜ W ( R ) k,n − ˜ W ( R ) k | ≥ ε o ∪ n | ˜ W ( R ) k − ˜ W ∞ | ≥ ε o ∪          log X | m |≤ C n 3 / 4 E ( ˜ Z m ) exp( θ m b X ( R ) ℓ,n − θ 2 m / 2) − log ˜ c n − log Z C − C exp y 2 p 2( d − 1) X ( R ) ℓ −  1 12 + 1 4( d − 1)  y 4 ! d y      ≥ ε ) ∪ n    log µ ( X ( R ) ℓ ) ([ − C, C ])    ≥ ε o ∪ { X ( R ) ℓ  = X } . This means that {| ∆ ′ n | ≥ 6 ε } ⊆ B n ∪ n | ˜ W ( R ) k,n − ˜ W ( R ) k | ≥ ε o ∪ n | ˜ W ( R ) k − ˜ W ∞ | ≥ ε o ∪ { X ( R ) ℓ  = X } , therefore, as we ha ve already explained, if k , R and n are large enough, P ( | ∆ ′ n | ≥ 6 ε ) ≤ 11 ε. This shows that indeed, ∆ ′ n P − − − → n →∞ 0 . 4 Momen t calculations Before explaining the calculations, we state and prov e the follo wing tec hnical lemma, whic h will b e useful throughout this section. Lemma 4.1. L et D ⊆ R d b e a c omp act set and f 1 , f 2 : D → R two c ontinuous functions with the fol lowing pr op erties: • f 1 has a unique glob al maximum at x ∈ D . • f 2 has a unique glob al minimum at x , with f 2 ( x ) = 0 . 24 • c 0 := 1 2 lim inf y → x y ∈ D f 1 ( x ) − f 1 ( y ) f 2 ( y ) > 0 . Then, for some c > 0 , f 1 ( y ) ≤ f 1 ( x ) − cf 2 ( y ) for any y ∈ D . Pr o of. Due to the third prop erty , there exists some δ > 0 suc h that f 1 ( x ) − f 1 ( y ) f 2 ( y ) ≥ c 0 ⇔ f 1 ( y ) ≤ f 1 ( x ) − c 0 f 2 ( y ) , for all y suc h that | y − x | ≤ δ . On the other hand, b ecause of the ﬁrst tw o prop erties, the con tinuit y of the functions and the compactness of D , there exists some ε > 0 suc h that f 1 ( y ) ≤ f 1 ( x ) − ε and f 2 ( y ) ≤ ε − 1 for an y y such that | y − x | ≥ δ . Therefore, for an y suc h y , f 1 ( y ) ≤ f 1 ( x ) − ε ≤ f 1 ( x ) − ε 2 f 2 ( y ) . The Lemma follows, if we set c = min( c 0 , ε 2 ) . Remark 4.2. W e wil l use this L emma in situations in which f 1 is C ∞ ar ound its maximizer and f 2 is p art of the T aylor exp ansion of f 1 ar ound the maximizer. 4.1 Regular Graphs The goal of this subsection is to pro ve Prop osition 3.2. F or the second moment calculation, w e will need the follo wing Prop osition, regarding the maximizer of Φ , when the vertex-empirical distribution ¯ h is ﬁxed. Prop osition 4.3. L et µ b e a spin system on a G ∼ G n,d , with a ﬁnite spin set X and weights ( ψ , ψ ) . Assume that ψ > 0 . F or a given vertex-empiric al distribution ¯ h such that ¯ h ( x ) > 0 for any x ∈ X , let B ¯ h = ( h ∈ [0 , 1] R X 2 : h ( x, y ) = h ( y , x ) ∀ x, y and X y ∈X h ( x, y ) = ¯ h ( x ) ∀ x ) and Φ ¯ h : B ¯ h → R b e the function Φ ¯ h ( h ) := H ( h ) + ⟨ h, log ψ ⟩ . Then, the maximizer of Φ ¯ h is an e dge-empiric al distribution h ∗ of the form h ∗ ( x, y ) ∝ q ( x ) q ( y ) ψ ( x, y ) , (13) for some q : X → (0 , ∞ ) such that h ∗ ∈ B h . Pr o of. Without loss of generality , w e may assume that P x,y ∈X ψ ( x, y ) = 1 . A t ﬁrst, we pro ve that a q such that the h ∗ deﬁned abov e is in B h exists. F or that purp ose, w e use Brou wer’s 25 ﬁxed p oin t Theorem. Without loss of generalit y , we will lo ok for q ∈ M 1 ( X ) . Deﬁne the function F : M 1 ( X ) → M 1 ( X ) to b e F ( q )( x ) = h ( x ) P y ∈X ψ ( x,y ) q ( y ) P z ∈X h ( z ) P y ∈X ψ ( z ,y ) q ( y ) . Then, b ecause X is ﬁnite and ψ > 0 , the denominators stay uniformly aw ay from 0 and, therefore, F is con tinuous. So, due to Brouw er’s ﬁxed p oin t Theorem, F has a ﬁxed p oin t q , whic h satisﬁes the conditions needed. Let q satisfy the conditions discussed ab o ve and write h ∗ ( x, y ) = q ( x ) q ( y ) ψ ( x, y ) /z for some z > 0 . Observe that H ( h ) + ⟨ h, log ψ ⟩ = − D KL ( h || ψ ) . Therefore, for any h ∈ B h , D KL ( h || ψ ) − D KL ( h || h ∗ ) = X x,y ∈X h ( x, y ) log h ∗ ( x, y ) ψ ( x, y ) = − log ( z ) + X x,y ∈X h ( x, y )(log q ( x ) + log q ( y )) = − log ( z ) + 2 · X x ∈X h ( x ) log q ( x ) . Observ e that this expression do es not dep end on h ∈ B ¯ h , so it turns out that D KL ( h || ψ ) = D KL ( h || h ∗ ) + D KL ( h ∗ || ψ ) ≥ D KL ( h ∗ || ψ ) , pro ving that indeed, h ∗ is the maximizer of Φ . Also, since D KL ( µ || ν ) = 0 ⇔ µ = ν , this maximizer is unique. 4.1.1 First moment calculation W e p erform the calculations proving Lemma 3.1 and the ﬁrst statement of Lemma 3.2. T o that end, we use (4) for the Ising mo del. A t the critical temp erature, we are still in the uniqueness regime, so the unique maximizer of Φ 1 is at h ∗ ( x, x ′ ) ∝ exp( β xx ′ ) . This means that h ∗ (1 , 1) = h ∗ ( − 1 , − 1) = d 4( d − 1) and h ∗ (1 , − 1) = h ∗ ( − 1 , 1) = d − 2 4( d − 1) . Set m ( h ) = n ( h (1 , 1) − h ( − 1 , − 1)) and s ( h ) = n ( h (1 , 1) + h ( − 1 , − 1)) . F or brevit y purp oses, whenever h is implied, we will just use s instead of s ( h ) . Let Φ 1 b e the function Φ corresp onding to the Ising mo del and H n,m = { h ∈ H n : m ( h ) = m } , H (1) n,m =  h ∈ H n,m :     s ( h ) − d 2( d − 1) · n     ≤ n 3 / 5  . 26 Around h ∗ , for h ∈ H n,m , Φ 1 ( h ) = Φ 1 ( h ∗ ) − ( d − 1) 2 d − 2 ·  s n − d 2( d − 1)  2 − ( d − 1)( d − 2)(3 d − 2) 12 d 2 ·  m n  4 + ( d − 1) 2 d ·  s n − d 2( d − 1)  ·  m n  2 + O     s n − d 2( d − 1)     3 +  m n  6 ! . Due to Lemma 4.1, there exists some constant c > 0 such that Φ 1 ( h ) ≤ Φ 1 ( h ∗ ) − c  s n − d 2( d − 1)  2 +  m n  4 ! (14) for every h . Pr o of of L emma 3.1. Let H (1) n b e the set of h such that ( s n − d 2( d − 1) ) 2 + ( m n ) 4 ≥ n − 4 / 5 . F or ev ery suc h h w e use the b ound in (14) to sho w that E ( Z h ) ≤ exp( − cn 1 / 5 ) · n Θ(1) exp( n Φ 1 ( h ∗ )) , whic h implies that X h ∈H (1) n E ( Z h ) ≤ exp( − cn 1 / 5 ) · n Θ(1) exp( n Φ 1 ( h ∗ )) . (15) Also, let H (2) n b e the set of h such that n − 4 / 5 > ( s n − d 2( d − 1) ) 2 + ( m n ) 4 ≥ C 4 n for some (large) constan t C > 0 . F or every h ∈ H (2) n , due to Stirling’s formula, E ( Z h ) ≍ n − 1 exp( n Φ 1 ( h )) , therefore X h ∈H (2) n E ( Z h ) ≲ n − 1 X h ∈H (2) n exp( n Φ 1 ( h )) ≤ n − 1 exp( n Φ 1 ( h ∗ )) X h ∈H (2) n exp " − cn  s n − d 2( d − 1)  2 +  m n  4 !# ≤ n − 1 exp( n Φ 1 ( h ∗ )) X k ≥ C 4 exp( − ck ) · |{ h : k n ≤ ( s n − d 2( d − 1) ) 2 + ( m n ) 4 < k +1 n }| ≲ n 1 / 4 exp( n Φ 1 ( h ∗ )) X k ≥ C 4 k 3 / 4 exp( − ck ) , (16) as |{ h : k n ≤ ( s n − d 2( d − 1) ) 2 + ( m n ) 4 < k +1 n }| = O ( n 5 / 4 · k 3 / 4 ) for any k ∈ N . Also, if H (3) n = { h : ( s n − d 2( d − 1) ) 2 + ( m n ) 4 ≤ 1 n } , then Φ 1 ( h ) ≥ Φ 1 ( h ∗ ) − c ′ n for any h ∈ H (3) n , therefore X | m |≤ C n 3 / 4 E ( Z m ) ≥ X h ∈H (3) n E ( Z h ) ≳ n − 1 ·   H (3) n   · exp( n Φ 1 ( h ∗ )) ≳ n 1 / 4 · exp( n Φ 1 ( h ∗ )) . This, together with (15), (16) and Mark ov’s inequalit y , implies the lemma. 27 Pr o of of ﬁrst statement of L emma 3.2. F or h ∈ H n,m \ H (1) n,m , Φ 1 ( h ) ≤ Φ( h ∗ ) − cn − 4 / 5 , whic h implies X h ∈H n,m \H (1) n,m E ( Z h ) ≤ exp( − cn 1 / 5 ) · n Θ(1) exp( n Φ 1 ( h ∗ )) . (17) On the other hand, for h ∈ H (1) n,m , due to Stirling’s formula, E ( Z h ) ∼ 2 5 / 2 ( d − 1) 3 / 2 π nd 3 / 2 ( d − 2) 1 / 2 · exp( n Φ 1 ( h )) . Therefore: X h ∈H (1) n,m E ( Z h ) ∼ 2 5 / 2 ( d − 1) 3 / 2 π nd 3 / 2 ( d − 2) 1 / 2 · X h ∈H (1) n,m exp( n Φ 1 ( h )) ∼ 2 1 / 2 ( d − 1) 3 / 2 π d 1 / 2 ( d − 2) 1 / 2 · Z R exp( n Φ 1 ( h )) d h ∼ 2 1 / 2 √ d − 1 √ n √ π d · exp( n Φ 1 ( h ∗ )) · exp  − ( d − 1)( d − 2) 12 d 2 · m 4 n 3  , where in the last step we used Laplace’s metho d. Since it is easy to c heck that Φ 1 ( h ∗ ) = log 2 + d 2 log(cosh( β )) , this ﬁnishes the ﬁrst moment calculation. 4.1.2 Second moment calculation Pr o of of the se c ond statement of L emma 3.2. W e need to show that E ( Z 2 m ) ∼ 2( d − 1) 3 / 2 nπ d √ d − 2 · (4 · cosh( β ) d ) n · exp  − ( d − 1)( d − 2) 6 d 2 · m 4 n 3  . (18) Observ e that on a graph G , Z 2 m = X σ,τ : m ( σ )= m ( τ )= m exp β X u ∼ v ( σ u σ v + τ u τ v ) ! . (19) Consider the tw o-copy mo del on G , i.e. the spin system on X = {− 1 , 1 } 2 with ψ (( x, x ′ ) , ( y , y ′ )) = exp( β ( xy + x ′ y ′ )) . F rom now on, for simplicity , w e will just write ( x, x ′ , y , y ′ ) instead of (( x, x ′ ) , ( y , y ′ )) , when appropriate. Then, due to (19), Z 2 m is the contribution to the partition function of all the edge-empirical distributions h for whic h both magnetizations are m , i.e. those satisfying the equations ¯ h (1 , 1) + ¯ h (1 , − 1) − ¯ h ( − 1 , 1) − ¯ h ( − 1 , − 1) = ¯ h (1 , 1) + ¯ h ( − 1 , 1) − ¯ h (1 , − 1) − ¯ h ( − 1 , − 1) = m n ⇔ ¯ h (1 , 1) − ¯ h ( − 1 , − 1) = m n and ¯ h (1 , − 1) = ¯ h ( − 1 , 1) . (20) 28 It is imp ortant to note that the unique maximizer h ∗ of the function Φ 2 that arises in this case is again the one for whic h h ( x, x ′ , y , y ′ ) ∝ exp( β ( xy + x ′ y ′ )) , i.e. h ∗ (1 , 1 , 1 , 1) = h ∗ (1 , − 1 , 1 , − 1) = h ∗ ( − 1 , 1 , − 1 , 1) = h ∗ ( − 1 , − 1 , − 1 , − 1) = d 2 16( d − 1) 2 h ∗ (1 , 1 , 1 , − 1) = h ∗ (1 , 1 , − 1 , 1) = h ∗ ( − 1 , − 1 , 1 , − 1) = h ∗ ( − 1 , − 1 , − 1 , 1) = d ( d − 2) 16( d − 1) 2 h ∗ (1 , 1 , − 1 , − 1) = h ∗ (1 , − 1 , − 1 , 1) = ( d − 2) 2 16( d − 1) 2 . It is also imp ortant to note that the corresp onding vertex-empirical distribution is the uni- form one, i.e. ¯ h ∗ (1 , 1) = ¯ h ∗ (1 , − 1) = ¯ h ∗ ( − 1 , 1) = ¯ h ∗ ( − 1 , − 1) = 1 / 4 . W e call ¯ H n,m ⊗ 2 the set of vertex-empirical distributions that satisfy the equalities (20) and ¯ H (1) n,m ⊗ 2 =  ¯ h ∈ ¯ H n,m ⊗ 2 :     t ( ¯ h ) − 1 2     ≤ n − 2 / 5  , where for a v ertex-empirical distribution ¯ h we set t ( ¯ h ) = ¯ h (1 , 1) + ¯ h ( − 1 , − 1) . Then, E ( Z 2 m ) = X ¯ h ∈ ¯ H n,m ⊗ 2 X h ∈ B ¯ h E ( Z h ) where B ¯ h w as deﬁned in Prop osition 4.3. F or any ¯ h ∈ ¯ H n,m ⊗ 2 , w e denote b y ¯ h ∗ the maximizer of the function Φ ¯ h , as explained in Prop osition 4.3. Using Prop osition 4.3, a calculation of the T a ylor expansion of Φ 2 ( ¯ h ∗ ) around h ∗ , veriﬁed using Mathematica yields: Φ 2 ( h ∗ ) = Φ 2 ( h ∗ ) − 2( d − 1)( d − 2) d 2 − 2 d + 2 ·  t − 1 2  2 + 2( d − 1)( d − 2) d 2 − 2 d + 2 ·  t − 1 2  ·  m n  2 − ( d − 2)( d − 1)(2 d 2 − d + 1) 3 d 2 ( d 2 − 2 d + 2) ·  m n  4 + O     t − 1 2     3 +  m n  6 ! . (21) F or an y ¯ h ∈ ¯ H n,m ⊗ 2 \ ¯ H (1) n,m ⊗ 2 , we can see that b ecause of Lemma 4.1, X h ∈ B ¯ h exp( n Φ 2 ( h )) ≤ | B ¯ h | · exp( n Φ 2 ( ¯ h ∗ )) ≤ n Θ(1) · exp( n Φ 2 ( ¯ h ∗ )) ≤ n Θ(1) · exp( n Φ 2 ( h ∗ )) · exp( − cn 1 / 5 ) , therefore X ¯ h ∈ ¯ H n,m ⊗ 2 \ ¯ H (1) n,m ⊗ 2 X h ∈ B ¯ h E ( Z h ) ≤ n Θ(1) · exp( n Φ 2 ( h ∗ )) · exp( − cn 1 / 5 ) . On the other hand, for every ¯ h ∈ ¯ H (1) n,m ⊗ 2 , ¯ h is within an O ( n − 1 / 4 ) distance from ¯ h ∗ . As a result, due to the diﬀerentiabilit y of Φ 2 the Hessians H ¯ h and H ¯ h ∗ of the functions Φ ¯ h and Φ ¯ h ∗ 29 also satisfy ∥ H ¯ h − H ¯ h ∗ ∥ OP = O ( n − 1 / 4 ) . A direct computation of the Hessian H ¯ h ∗ , veriﬁed b y Mathematica yields: det( − H ¯ h ∗ ) = 2 29 · ( d − 1) 16 · ( d 2 − 2 d + 2) ( d − 2) 8 · d 4 > 0 , (22) whic h means that on a region of ¯ h ∗ , H ¯ h is negativ e-deﬁnite and b ounded a w ay from a singular matrix. In other words, k eeping in mind Lemma 4.1 as well, there exists a constan t c > 0 for which Φ ¯ h ( h ) ≤ Φ 2 ( ¯ h ∗ ) − c ∥ h − ¯ h ∗ ∥ 2 2 . This, working in the same wa y as b efore, implies that if w e set H (1) n, ¯ h =  h ∈ B ¯ h : ∥ h − ¯ h ∗ ∥ 2 ≤ n − 2 / 5  , then X h ∈ B ¯ h \H (1) n, ¯ h E ( Z h ) ≤ n Θ(1) · exp( n Φ 2 ( ¯ h ∗ )) · exp( − cn 1 / 5 ) . If for some h , ¯ h ∈ ¯ H (1) n,m ⊗ 2 and h ∈ H (1) n, ¯ h , due to (4) and Stirling’s form ula, E ( Z h ) ∼ 2 17 · ( d − 1) 10 ( π n ) 9 / 2 · d 9 · ( d − 2) 4 · exp( n Φ 2 ( h )) , so X ¯ h ∈ ¯ H (1) n,m ⊗ 2 X h ∈H (1) n, ¯ h E ( Z h ) ∼ 2 17 · ( d − 1) 10 ( π n ) 9 / 2 · d 9 · ( d − 2) 4 X ¯ h ∈ ¯ H (1) n,m ⊗ 2 X h ∈H (1) n, ¯ h exp( n Φ 2 ( h )) (23) W e claim that all the sums of the form exp( − n Φ 2 ( ¯ h ∗ )) X h ∈H (1) n, ¯ h exp( n Φ 2 ( h )) , for ¯ h ∈ ¯ H (1) n,m ⊗ 2 , are within a 1 + o (1) factor. Indeed, for any such ¯ h and h ∈ H (1) n, ¯ h , Φ 2 ( h ) = Φ 2 ( ¯ h ∗ ) + 1 2 ⟨ h − ¯ h ∗ , H ¯ h ( h − ¯ h ∗ ) ⟩ + o ( n − 1 ) , therefore exp( − n Φ 2 ( ¯ h ∗ )) X h ∈H (1) n, ¯ h exp( n Φ 2 ( h )) ∼ X h ∈H (1) n, ¯ h exp  n 2 ⟨ h − ¯ h ∗ , H ¯ h ( h − ¯ h ∗ ) ⟩  . Also, since ∥ H ¯ h − H ¯ h ∗ ∥ OP = O ( n − 1 / 4 ) as explained ab o ve, for an y h ∈ H (1) n, ¯ h , ⟨ h − ¯ h ∗ , H ¯ h ( h − ¯ h ∗ ) ⟩ = ⟨ h − ¯ h ∗ , H ¯ h ∗ ( h − ¯ h ∗ ) ⟩ + ⟨ h − ¯ h ∗ , ( H ¯ h − H ¯ h ∗ )( h − ¯ h ∗ ) ⟩ = ⟨ h − ¯ h ∗ , H ¯ h ∗ ( h − ¯ h ∗ ) ⟩ + o ( n − 1 ) , 30 b ecause |⟨ h − ¯ h ∗ , ( H ¯ h − H ¯ h ∗ )( h − ¯ h ∗ ) ⟩| ≤ ∥ h − ¯ h ∗ ∥ 2 2 · ∥ H ¯ h − H ¯ h ∗ ∥ OP = O ( n − 4 / 5 − 1 / 4 ) = o ( n − 1 ) . Putting all of this together implies exp( − n Φ 2 ( ¯ h ∗ )) X h ∈H (1) n, ¯ h exp( n Φ 2 ( h )) ∼ X h ∈H (1) n, ¯ h exp  − n 2 ⟨ h − ¯ h ∗ , H ¯ h ( h − ¯ h ∗ ) ⟩  ∼ X h ∈H (1) n, ¯ h exp  − n 2 ⟨ h − ¯ h ∗ , H ¯ h ∗ ( h − ¯ h ∗ ) ⟩  ∼ exp( − n Φ 2 ( h ∗ )) X h ∈H (1) n, ¯ h ∗ exp( n Φ 2 ( h )) ∼ d 8 · ( d − 2) 4 · ( π n ) 3 2 29 / 2 · ( d − 1) 8 · √ d 2 − 2 d + 2 , where the last equality follows b ecause of (22) and Laplace’s metho d. W e combine ev erything to prov e (18). W e w ork as in the calculation of the ﬁrst momen t. X ¯ h ∈ ¯ H (1) n,m ⊗ 2 X h ∈H (1) n, ¯ h exp( n Φ 2 ( h )) = X ¯ h ∈ ¯ H (1) n,m ⊗ 2 exp( n Φ 2 ( ¯ h ∗ )) ·    exp( − n Φ 2 ( ¯ h ∗ )) X h ∈H (1) n, ¯ h exp( n Φ 2 ( h ))    ∼ d 8 · ( d − 2) 4 · ( π n ) 3 2 29 / 2 · ( d − 1) 8 √ d 2 − 2 d + 2 X ¯ h ∈ ¯ H (1) n,m ⊗ 2 exp( n Φ 2 ( ¯ h ∗ )) ∼ d 8 ( d − 2) 7 / 2 ( π n ) 7 / 2 2 16 · ( d − 1) 17 / 2 · exp  n Φ 2 ( h ∗ ) − ( d − 1)( d − 2) m 4 6 d 2 n 3  , where the last equality is due to Laplace’s metho d. T ogether with (23) and (19), this implies E ( Z 2 m ) ∼ 2 · ( d − 1) 3 / 2 π nd · √ d − 2 · exp( n Φ 2 ( h ∗ )) · exp  − ( d − 1)( d − 2) 6 d 2 · m 4 n 3  , and since Φ 2 ( h ∗ ) = log (4) + d log cosh( β ) , w e ha ve successfully pro ven (18). 4.2 Erdös-Rén yi Graphs 4.2.1 First moment calculation F or an y σ , ˜ Z σ is a pro duct of indep endent random v ariables and its exp ectation only dep ends on m ( σ ) . T rivially , the n um b er of u suc h that σ u = 1 is n + m 2 , while the n umber of u suc h that σ u = − 1 is n − m 2 . Moreov er, the n umber of pairs { u, v } such that σ u σ v = 1 is  n + m 2 2  +  n − m 2 2  = n 2 + m 2 4 − n 2 , whereas the n umber of { u, v } suc h that σ u σ v = − 1 is 31 n + m 2 · n − m 2 = n 2 − m 2 4 . Therefore, if m ( σ ) = m , then since e ± β cosh( β ) − 1 = ± 1 d , E ( ˜ Z σ ) = 2 − n  1 + 1 n  n 2 + m 2 4 − n 2 ·  1 − 1 n  n 2 − m 2 4 ∼ 2 − n · exp  m 2 2 n − 3 4  , as log (1 ± 1 n ) = ± 1 n − 1 2 n 2 + O ( n − 3 ) . T aylor expanding the function H ( x ) = − 1+ x 2 log(1 + x ) − 1 − x 2 log(1 − x ) around x = 0 implies that E ( ˜ Z m ) =  n n + m 2  · E ( ˜ Z σ ) ∼ √ 2 √ π n · exp  H  m n  + m 2 2 n − 3 4  ∼ √ 2 √ π n · exp  − m 4 12 n 3 − 3 4  . Therefore, X | m |≤ n 3 / 4 E ( ˜ Z m ) ≳ n 1 / 4 and for any C > 0 , X | m |≤ C n 3 / 4 E ( ˜ Z m ) ≲ n − 1 / 2 C X k =0 X k ≤| m |· n − 3 / 4 ≤ k +1 exp( − m 4 12 n 3 ) ≲ n 1 / 4 C X k =0 exp( − k 4 12 ) ≲ n 1 / 4 . It follows that indeed, ˜ c 0 > 0 . Also, b ecause of Lemma 4.1, for | m | ≥ C n 3 / 4 , E ( ˜ Z m ) ≲ n − 1 / 2 · exp( − c m 4 n 3 ) . This implies X | m | >C n 3 / 4 E ( ˜ Z m ) ≲ X k ≥ C X k ≤| m |· n − 3 / 4 ≤ k +1 n − 1 / 2 exp( − c m 4 n 3 ) ≲ n 1 / 4 · X k ≥ C exp( − ck 4 ) . Cho osing C > 0 to b e large enough ﬁnishes the pro of of Lemma 3.6. The ﬁrst statemen t of Prop osition 3.7 is already pro v en. 4.2.2 Second moment calculation Pr o of of the se c ond statement of Pr op osition 3.7. F or any σ, σ ′ ∈ {− 1 , 1 } n with m ( σ ) = m ( σ ′ ) = m , set t ( σ, σ ′ ) = 1 √ n · X u σ u σ ′ u − m 2 n ! . F rom no w on, for simplicit y purp oses, if σ, σ ′ are ﬁxed, we will just use t instead of t ( σ, σ ′ ) . By the deﬁnition of t , the n umber of v ertices u such that: 32 • ( σ u , σ ′ u ) = (1 , 1) is n ˜ h (1 , 1) = n + √ nt + m 2 n +2 m 4 . • ( σ u , σ ′ u ) = ( − 1 , − 1) is n ˜ h ( − 1 , − 1) = n + √ nt + m 2 n − 2 m 4 . • ( σ u , σ ′ u ) = (1 , − 1) is n ˜ h (1 , − 1) = n − √ nt − m 2 n 4 . • ( σ u , σ ′ u ) = ( − 1 , 1) is n ˜ h ( − 1 , 1) = n − √ nt − m 2 n 4 . This means that the num b er of pairs { u, v } such that: • σ u σ v + σ ′ u σ ′ v = 2 is  n + √ nt + m 2 n +2 m 4 2  +  n + √ nt + m 2 n − 2 m 4 2  + 2 ·  n − √ nt − m 2 n 4 2  = n 2 + ( √ nt + m 2 n ) 2 8 + m 2 4 − n 2 . • σ u σ v + σ ′ u σ ′ v = 0 is n + √ nt + m 2 n + 2 m 4 + n + √ nt + m 2 n − 2 m 4 ! · n − √ nt − m 2 n 4 + n − √ nt − m 2 n 4 ! = n 2 − ( √ nt + m 2 n ) 2 4 • σ u σ v + σ ′ u σ ′ v = − 2 is n + √ nt + m 2 n + 2 m 4 · n + √ nt + m 2 n − 2 m 4 + n − √ nt − m 2 n 4 ! 2 = n 2 + ( √ nt + m 2 n ) 2 8 − m 2 4 . Keeping all of this in mind and com bining it with the facts that e ± 2 β cosh( β ) 2 − 1 = ± 2 d + 1 d 2 and 1 cosh( β ) 2 − 1 = − 1 d 2 implies that E ( ˜ Z σ ˜ Z σ ′ ) ∼ 4 − n  1 + 2 d +1 dn  n 2 +( √ nt + m 2 n ) 2 8 + m 2 4 − n 2  1 − 1 dn  n 2 − ( √ nt + m 2 n ) 2 4  1 − 2 d − 1 dn  n 2 +( √ nt + m 2 n ) 2 8 − m 2 4 ∼ 4 − n · exp   2 d +1 dn − (2 d +1) 2 2 d 2 n 2   n 2 +( √ nt + m 2 n ) 2 8 + m 2 4 − n 2  +  − 1 dn − 1 2 d 2 n 2  · n 2 − ( √ nt + m 2 n ) 2 4 +  − 2 d − 1 dn − (2 d − 1) 2 2 d 2 n 2   n 2 +( √ nt + m 2 n ) 2 8 − m 2 4  # ∼ 4 − n · exp  m 2 n + t 2 2 d + m 2 t dn 3 / 2 + m 4 2 dn 3 − 1+2 d +6 d 2 4 d 2  . 33 F or an y t , let ˜ Z m,t = X ( σ,σ ′ ): m ( σ )= m ( σ ′ )= m t ( σ,σ ′ )= t ˜ Z σ ˜ Z σ ′ . F or an y t , n !  n ˜ h (1 , 1)  ! ·  n ˜ h ( − 1 , − 1)  ! ·  n ˜ h (1 , − 1)  ! 2 ≤ n Θ(1) · exp( nH ( ˜ h )) , where H ( ˜ h ) is the Shannon entrop y of the probabilit y distribution ˜ h . So, E ( ˜ Z m,t ) ≤ n Θ(1) · exp h n  log(4) + m 2 n 2 + t 2 2 dn + m 2 t dn 5 / 2 + H ( ˜ h ) i = n Θ(1) · exp  ˜ Φ( m n , t √ n )  , (24) where ˜ Φ is the function ˜ Φ( x, y ) = x 2 + y 2 2 d + x 2 y d − (1 + x ) 2 + y 4 · log  (1 + x ) 2 + y  − (1 − x ) 2 + y 4 · log  (1 − x ) 2 + y  − 1 − x 2 − y 2 · log  1 − x 2 − y  . Around (0,0), the T aylor expansion of ˜ Φ is ˜ Φ( x, y ) = − d − 1 2 d · y 2 −  1 6 − 1 2 d  · x 4 + 1 d · x 2 y + O  y 3 + x 6  , so there exists some constant c > 0 for whic h ˜ Φ( x, y ) ≤ − cy 2 + O ( y 3 + x 4 ) . Viewing ˜ Φ as a function of y , with ﬁxed x Lemma 4.1 implies that n ˜ Φ( m n , t √ n ) ≤ − ct 2 + O ( m 4 n 3 ) . Therefore, for | t | > n 1 / 10 , by plugging this back into (24) w e ﬁnd that E ( ˜ Z m,t ) ≤ n Θ(1) · exp  − cn 1 / 5  . (25) On the other hand, for any | t | ≤ n 1 / 10 , the p ossible c hoices for σ and σ ′ are n !  n + √ nt + m 2 n +2 m 4  ! ·  n + √ nt + m 2 n − 2 m 4  ! ·  n − √ nt − m 2 n 4  ! 2 ∼ 4 n 4 √ 2 ( π n ) 3 / 2 exp  − m 2 n − t 2 2 − m 4 6 n 3  . So, keeping (25) in mind, E ( ˜ Z 2 m ) ∼ 4 √ 2 ( π n ) 3 / 2 · exp  m 4 2 dn 3 − m 4 6 n 3 − 1 + 2 d + 6 d 2 4 d 2  · X | t |≤ n 1 / 10 exp  − d − 1 2 d t 2 + m 2 dn 3 / 2 t  ∼ 2 π n · exp  − m 4 6 n 3 + m 4 2( d − 1) n 3 − 1 + 2 d + 6 d 2 4 d 2  · r d d − 1 . This prov es the second statemen t of Prop osition 3.7. 34 5 Cycle and path coun ts 5.1 Cycle coun ts in G n,d The goal of this subsection is to prov e Prop osition 3.3. In fact, since statement 1 is very w ell-known, w e only prov e statemen t 2. The analog of this statement w as pro ven in [10] for the an ti-ferromagnetic Ising mo del, how ever there are some minor diﬀerences with our case, whic h is wh y w e pro ve it here. F or an y h ∈ H (1) n,m and i ≥ 1 , we prov e that E ( Z h · Y i,n ) E ( Z h ) ∼ 1 + ( d − 1) − i . (26) F or any C of length i that has vertices v 1 , . . . , v i , we ﬁrst plant σ C ∈ {− 1 , 1 } C , which also induces the num b ers k ++ , k + − and k −− of edges in C that connect t wo vertices with +1 ’s, one with +1 and one with − 1 and t wo with − 1 ’s, resp ectively . W e now construct the rest of the graph, in exactly the same w ay as in (3). There are  n − i n − i + m − m ( σ C ) 2  w ays to put the remaining spins on the v ertices,  d n + m 2 − 2 k ++ − k + − dnh (1 , 1) − 2 k ++  ·  d n − m 2 − 2 k −− − k + − dnh ( − 1 , − 1) − 2 k −−  w ays to put the spins on the half-edges and ( dnh (1 , 1) − 2 k ++ − 1)!! · ( dnh ( − 1 , − 1) − 2 k −− − 1)!! · ( dnh (1 , − 1) − k + − )! w ays to connect the half edges in an admissible wa y . Therefore, for any C , E ( Z h | C ∈ G ) = 1 ( dn − 2 i − 1)!! X σ C  n − i n − i + m − m ( σ C ) 2  d n + m 2 − 2 k ++ − k + − dnh (1 , 1) − 2 k ++  ·  d n − m 2 − 2 k −− − k + − dnh ( − 1 , − 1) − 2 k −−  · ( dnh (1 , 1) − 2 k ++ − 1)!! · ( dnh ( − 1 , − 1) − 2 k −− − 1)!! · ( dnh (1 , − 1) − k + − )! Using (3), Stirling’s form ula and the fact that for h ∈ H (1) n,m , h − h ∗ = o (1) , w e ﬁnd that E ( Z h | C ∈ G ) E ( Z h ) ∼  dn 2  i X σ C 4 i ( dn ) − i h (1 , 1) k ++ h ( − 1 , − 1) k −− h (1 , − 1) k + − ∼ X σ C (2 h (1 , 1)) k ++ (2 h ( − 1 , − 1)) k −− (2 h (1 , − 1)) k + − ∼ X σ C 1 (2 cosh( β )) i exp β i X j =1 σ j σ j +1 ! ∼ 1 + ( d − 1) − i , whic h pro ves (26). As w e already explained in (17), E    Y i,n X h ∈H n,m \H (1) n,m Z h    ≤ n Θ(1) · X h ∈H n,m \H (1) n,m E ( Z h ) ≤ exp( − cn 1 / 5 ) · n Θ(1) · E ( Z m ) = o ( E ( Z m )) . 35 Com bining this with (26), w e get that E ( Z m · Y i,n ) E ( Z m ) = P C ∈ G P h ∈H n,m E ( Z h 1 C ∈ G ) P h ∈H n,m E ( Z h ) ∼ ( d − 1) i 2 i P h ∈H (1) n,m E ( Z h | C ∈ G ) P h ∈H (1) n,m E ( Z h ) ∼ ( d − 1) i + 1 2 i . Therefore, the Y i,n ha ve the correct ﬁrst momen ts under the planted measure. Proving the asymptotic indep endence, as w ell as the distribution of the limit relies on calculating the join t factorial momen ts of ( Y i,n ) under the planted measure. As in the simple case we ha ve demonstrated ab o ve, this comes do wn to sho wing that sev eral sequences of v ertices form cycles concurrently . Under the plan ted measure, the existence of short cycles with non- empt y ov erlap is of order O (1 /n ) . Therefore, the joint factorial moment calculation comes do wn to a sum ov er distinct vertices, and this is a trivial generalization of the calculation w e demonstated ab o ve. 5.2 Join t path and cycle coun ts in G ( n, d/n ) The goal of this subsection is to pro ve Prop osition 3.8. The ﬁrst step to wards that will b e to calculate the exp ectation and the v ariance of X ℓ,n , sho wing that the normalization p erformed in (2) makes sense. Lemma 5.1. 1. The r andom variable X ℓ,n has E ( X ℓ,n ) = (1 + O ( n − 1 )) · 1 2 nd ℓ and V ar ( X ℓ,n ) = (1 + O ( n − 1 )) · 1 2 nd 2 ℓ ℓ 2 d − 1  1 + γ (2) ℓ  , wher e lim ℓ →∞ γ (2) ℓ = 0 . 2. L et m b e as in statement 4 of Pr op osition 3.8. Then, under the plante d me asur e P ∗ m , E ∗ m ( X ℓ,n ) = (1 + O ( n − 1 )) · 1 2 nd ℓ  1 + m 2 n − 2 ℓ d − 1  1 + γ (1) ℓ   and V ar ∗ m ( X ℓ,n ) ∼ V ar ( X ℓ,n ) , wher e lim ℓ →∞ γ (1) ℓ = 0 . Pr o of. F or an y ℓ ∈ N , let  P ℓ = { ( v 0 , v 1 , . . . , v ℓ ) : v i ∈ [ n ] , v i  = v j ∀ i, j } b e the set of directed paths of length ℓ in the complete graph K n and P ℓ =  P ℓ / ∼ , where for tw o paths p (1)  = p (2) ∈  P ℓ w e write p (1) ∼ p (2) if v (1) i = v (2) ℓ − i for an y 0 ≤ i ≤ ℓ . P ℓ is the set of paths of length ℓ in the complete graph K n . Then, E ( X ℓ,n ) = X P ∈P ℓ P ( P ∈ G ) = |P ℓ | ·  d n  ℓ = (1 + O ( n − 1 )) · 1 2 nd ℓ , 36 as |P ℓ | = |  P ℓ | 2 = 1 2 n ( n − 1) · · · ( n − ℓ ) = ( 1 2 + O ( n − 1 )) · n ℓ +1 . Also, V ar ( X ℓ,n ) = X ( P,Q ) ∈P 2 ℓ E ( P ∩ Q )  = ∅ [ P ( P , Q ∈ G ) − P ( P ∈ G ) · P ( Q ∈ G )] = (1 + O ( n − 1 )) · ℓ X k =1 X ( P,Q ) ∈P 2 ℓ | E ( P ∩ Q ) | = k  d n  2 ℓ − k = (1 + O ( n − 1 )) · ℓ X k =1 |{ ( P , Q ) ∈ P ℓ : | E ( P ∩ Q ) | = k }| ·  d n  2 ℓ − k . W e now prov e that |{ ( P , Q ) ∈ P 2 ℓ : | E ( P ∩ Q ) | = k }| = (1 + O ( n − 1 )) · 1 2 ( ℓ − k + 1) 2 · n 2 ℓ − k +1 . First of all, observ e that the num b er of pairs of paths meeting in at least t wo diﬀerent segmen ts is O ( n 2 ℓ − k ) . Indeed, since at least k + 2 v ertices are common, there are at most 2 ℓ + 2 − ( k + 2) = 2 ℓ − k choices of new v ertices to b e made. So, most pairs of paths in the set { ( P , Q ) ∈ P 2 ℓ : | E ( P ∩ Q ) | = k } in tersect in a single path of length k . Note that if w e ask for |{ (  P ,  Q ) ∈  P 2 ℓ : | E (  P ∩  Q ) | = k }| instead, this cardinalit y is (1 + O ( n − 1 )) · ( ℓ − k + 1) 2 · n 2 ℓ − k +1 . Indeed, for ev ery c hoice of  P , there are ℓ − k + 1 w ays to choose the segment of  P that will b e common with  Q , another ℓ − k + 1 w ays to choose which part of  Q will cov er the c hosen part and another (1 + O ( n − 1 ) · n ℓ − k w ays to complete  Q . Ev ery path  P ∈  P ℓ is a c hoice of P ∈ P ℓ and a c hoice of a direction. F or any ( P , Q ) that intersect in a path of length k , there are tw o wa ys of choosing the direction of P and then exactly one w ay of choosing the direction of Q , as their common part already has a direction. Therefore, |{ ( P , Q ) ∈ P 2 ℓ : | E ( P ∩ Q ) | = k }| = (1 + O ( n − 1 )) · 1 2 ( ℓ − k + 1) 2 · n 2 ℓ − k +1 , as we initially claimed. Th us: V ar ( X ℓ,n ) =  1 2 + O ( n − 1 )  · nd 2 ℓ ℓ X k =1 ( ℓ − k + 1) 2 · d − k = (1 + O ( n − 1 )) · 1 2 nd 2 ℓ ℓ 2 d − 1  1 + γ (2) ℓ  , whic h pro ves the desired result. W e mov e on to the planted measure case. F or an y P = ( w 0 , w 1 , . . . , w ℓ ) ∈ P ℓ and σ suc h that m ( σ ) = m , E ( ˜ Z σ | P ∈ G ) E ( ˜ Z σ ) = exp β ℓ − 1 P i =0 σ w i σ w i +1 ! cosh( β ) ℓ ℓ − 1 Q i =0  1 + σ w i σ w i +1 n  = (1 + O ( n − 1 )) · exp  β ℓ − 1 P i =0 σ w i σ w i +1  cosh( β ) ℓ . 37 W e calculate E ∗ m ( X ℓ,n ) . E ∗ m ( X ℓ,n ) = E ( ˜ Z m X ℓ,n ) E ( ˜ Z m ) = 1 E ( ˜ Z m ) · X P ∈P ℓ E ( ˜ Z m 1 P ∈ G ) = 1 E ( ˜ Z m ) · X P ∈P ℓ P ( P ∈ G ) X σ : m ( σ )= m E ( ˜ Z σ | P ∈ G ) = (1 + O ( n − 1 )) 1 2 nd ℓ  n n + m 2  − 1 X σ : m ( σ )= m exp  β ℓ − 1 P i =0 σ w i σ w i +1  cosh( β ) ℓ = (1 + O ( n − 1 )) 1 2 nd ℓ  n n + m 2  − 1 X σ P ∈{− 1 , 1 } V ( P )  n − ℓ − 1 n − ℓ − 1+ m − m ( σ P ) 2  exp  β ℓ − 1 P i =0 σ w i σ w i +1  cosh( β ) ℓ = (1 + O ( n − 1 )) 1 2 nd ℓ X σ P ∈{− 1 , 1 } V ( P ) exp  β ℓ − 1 P i =0 σ w i σ w i +1  cosh( β ) ℓ · ℓ Y i =0 1 + σ w i m n 2 = (1 + O ( n − 1 )) 1 2 nd ℓ · E µ ℓ " ℓ Y i =0  1 + σ w i m n  # , where µ ℓ is the Ising mo del on the segment of length ℓ , at inv erse temp erature β = tanh − 1 ( d − 1 ) . Therefore, E ∗ m ( X ℓ,n ) = (1 + O ( n − 1 )) 1 2 nd ℓ · E µ ℓ " ℓ Y i =0  1 + σ w i m n  # = (1 + O ( n − 1 )) 1 2 nd ℓ · 1 + m 2 n 2 X 0 ≤ i 1 r 1 / 2 . (31) First, observe that every F i , as a subgraph of G i , has v ( F i ) ≥ e ( F i ) , and if i > r − r 1 and F i  = ∅ , F i is a forest, so v ( F i ) ≥ e ( F i ) + 1 . If L falls in to category 3, observe that r X i = r − r 1 +1 ( v ( F i ) − e ( F i )) = r 1 / 2 . Let j ∈ [ r − r 1 ] b e the smallest j for which F j  = ∅ . W e claim that F j is a forest. Indeed, since the ( G i ) r − r 1 i =1 are pairwise distinct, F j = G j can only happ en if G j connects to t wo G i 1 , G i 2 with i 1 < i 2 < j and V ( G i 1 ) ∩ V ( G i 2 )  = ∅ , contradicting the minimalit y of j . Since F j is a non-empty forest, v ( F j ) > e ( F j ) , concluding the pro of of (31). Supp ose L falls in to category 2. W e will sho w that r X i = r − r 1 +1 ( v ( F i ) − e ( F i )) > r 1 / 2 . Let S b e the set of i ∈ [ r − r 1 + 1 , r ] such that F i  = ∅ , c ( L ′ ) b e the num b er of connected comp onen ts of L ′ and n ( L ′ ) be the num b er of isolated p oin ts of L ′ . An y i for whic h G i is an isolated p oint of L ′ is in S , as the G i m ust ha ve a neighbor in the G 1 , . . . , G r − r 1 . Also, in the rest c ( L ′ ) − n ( L ′ ) connected comp onen ts that contain at least tw o v ertices, at most one v ertex p er connected comp onen t is not in S . W e observ e that 2 c ( L ′ ) − n ( L ′ ) ≤ r 1 , therefore r X i = r − r 1 +1 ( v ( F i ) − e ( F i )) ≥ | S | ≥ n ( L ′ ) + r 1 − ( c ( L ′ ) − n ( L ′ )) ≥ r 1 / 2 + 3 n ( L ′ ) / 2 . If n ( L ′ ) ≥ 1 , then w e hav e pro ven the desired result. Otherwise, n ( L ′ ) = 0 and c ( L ′ ) = r 1 / 2 , so L ′ is a p erfect matc hing. Due to the fact that L is in category 2, there exists a j ∈ S such that G j is connected to one of the G 1 , G 2 , . . . , G r − r 1 . Pick the smallest suc h j . Assume G j is connected to G i for some 1 ≤ i ≤ r − r 1 and to G j ′ , for some j ′ > r − r 1 . If j > j ′ , then F j is a forest with at least tw o connected comp onen ts, one formed b ecause of the in tersection with G i and one b ecause of the intersection with G j ′ (due to the minimality of j ′ , G j do es not connect to any v ertex in ( G i ) r − r 1 i =1 . Therefore, v ( F j ) − e ( F j ) ≥ 2 and no w (31) is prov en. On the other hand, if j < j ′ , then G j is the vertex app earing ﬁrst in its connected comp onent 42 in L ′ and is still in S . Therefore, | S | ≥ 1 + c ( L ′ ) > r 1 / 2 and (31) follo ws in this case as w ell. Since there are only O (1) num b er of choices for L and ( F i ) i , (29) has b een pro v en. Lemma 5.4. F or the class L 0 we have |L 0 | = ( r 1 − 1)!! and for any L 0 ∈ L 0 , X G 1 ,...,G r L = L 0 T ( G 1 , . . . , G r ) = (1 + o (1)) · V ar ( X ℓ,n ) r 1 / 2 · s Y i =3 ˜ λ r i i . (32) Under the plante d me asur e, the r elation is the same exc ept the ˜ λ i b e c ome ˜ µ i . Pr o of of L emma 5.4. The fact that |L 0 | = ( r 1 − 1)!! is easy to see, as there are exactly ( r 1 − 1)!! matc hings of r 1 v ertices. If L = L 0 , then T ( G 1 , . . . , G r ) = r − r 1 Y j =1 P ( G j ∈ G ) · r Y j = r − r 1 +1 q Co v ( 1 G j , 1 G v ( j ) ) , (33) where for an y j , v ( j ) is the unique vertex in L 0 for whic h G j ∩ G v ( j )  = ∅ . Fix G 1 , . . . , G r − r 1 whic h are pairwise disjoin t and let W = V ( G 1 ∪ · · · ∪ G r − r 1 ) , w = s P i =3 ir i b e the set and the total num b er of vertices used b y one of the G 1 , . . . , G r − r 1 . W e calculate: V ar ( X ℓ,n ) r 1 / 2 ∼ V ar ( X ℓ,n − w ) r 1 / 2 = X G r − r 1 +1 ,...,G r G r − r 1 + j ∩ W = ∅ G r − r 1 +2 j − 1 ∩ G r − r 1 +2 j  = ∅ ∀ j r 1 / 2 Y j =1 Co v ( 1 G r − r 1 +2 j − 1 , 1 G r − r 1 +2 j ) = X G r − r 1 +1 ,...,G r G j ∩ W = ∅ G j ∩ G v ( j )  = ∅ ∀ j r Y j = r − r 1 +1 q Co v ( 1 G j , 1 G v ( j ) ) = X G r − r 1 +1 ,...,G r L = L 0 r Y j = r − r 1 +1 q Co v ( 1 G j , 1 G v ( j ) ) + X G r − r 1 +1 ,...,G r G j ∩ G v ( j )  = ∅ ∀ j L  = L 0 r Y j = r − r 1 +1 q Co v ( 1 G j , 1 G v ( j ) ) . (34) When G j ∩ G v ( j )  = ∅ for all j and L  = L 0 , we kno w that L / ∈ L 0 . This case w as handled in Lemma 5.3, in the sp ecial case r 3 = · · · = r s = 0 : At ﬁrst, observe that      r Y j = r − r 1 +1 q Co v ( 1 G j , 1 G v ( j ) )      ≤ 2 r 1 · E " r Y j = r − r 1 +1 1 G j # 43 and then we pro ceed as we did once we prov ed (30) to prov e that X G r − r 1 +1 ,...,G r G j ∩ G v ( j )  = ∅ ∀ j L  = L 0 r Y j = r − r 1 +1 q Co v ( 1 G j , 1 G v ( j ) ) = o ( n r 1 / 2 ) = o ( V ar ( X ℓ,n ) r 1 / 2 ) . Com bining this with (34) it follo ws that for an y G 1 , . . . , G r − r 1 , X G r − r 1 +1 ,...,G r L = L 0 T ( G 1 , . . . , G r ) ∼ V ar ( X ℓ,n ) r 1 / 2 · r − r 1 Y j =1 P ( G j ∈ G ) ∼ V ar ( X ℓ,n ) r 1 / 2 ·  d n  w Because of Lemma 5.2, under the plan ted measure, w e ma y write X G r − r 1 +1 ,...,G r L = L 0 T ∗ m ( G 1 , . . . , G r ) ∼ V ar ( X ℓ,n ) r 1 / 2  d n  w s Y i =3 (1 + d − i ) r i . 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Distribution of the magnetization of the critical Ising model on sparse random graphs

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