Poisson-Dirichlet Statistics for the extremes of the two-dimensional discrete Gaussian Free Field

In a previous paper, the authors introduced an approach to prove that the statistics of the extremes of a log-correlated Gaussian field converge to a Poisson-Dirichlet variable at the level of the Gibbs measure at low temperature and under suitable test functions. The method is based on showing that the model admits a one-step replica symmetry breaking in spin glass terminology. This implies Poisson-Dirichlet statistics by general spin glass arguments. In this note, this approach is used to prove Poisson-Dirichlet statistics for the two-dimensional discrete Gaussian free field, where boundary effects demand a more delicate analysis.

where (S k , k ≥ 0) is a simple random walk with S 0 = v of law P v killed at the first exit time of A, τ A , i.e. the first time where the walk reaches the boundary ∂A.Throughout the paper, for any A ⊂ Z 2 , ∂A will denote the set of vertices in A c that share an edge with a vertex of A. We will write P for the law of the Gaussian field and E for the expectation.For B ⊂ A, we denote the σ-algebra generated by {φ v , v ∈ B} by F B .
We are interested in the case where A = V N := {1, . . ., N } 2 in the limit N → ∞.For 0 ≤ δ < 1/2, we denote by V δ N the set of the points of V N whose distance to the boundary ∂V N is greater than δN .In this set, the variance of the field diverges logarithmically with N , cf.Lemma 5.2 in the appendix, , where O N (1) will always be a term which is uniformly bounded in N and in v ∈ V N .(The term o N (1) will denote throughout a term which goes to 0 as N → ∞ uniformly in all other parameters.)Equation (1.2) follows from the fact that for v ∈ V δ N and , where • denotes the Euclidean norm on Z 2 .A similar estimate yields an estimate on the covariance In view of (1.2) and (1.3), the Gaussian field (φ v , v ∈ V N ) is said to be log-correlated.
On the other hand, there are many points that are outside V δ N (of the order of N 2 points) for which the estimates (1.2) and (1.3) are not correct.Essentially, the closer the points are to the boundary the lesser are the variance and covariance as the simple random walk in (1.1) has a higher probability of exiting V N early.This decoupling effect close to the boundary complicates the analysis of the extrema of the GFF by comparison with log-correlated Gaussian fields with stationary distribution.1.2.Main results.It was shown by Bolthausen, Deuschel, and Giacomin [7] that the maximum of the GFF in V δ N satisfies (1.4) lim , in probability.
A comparison argument using Slepian's lemma can be used to extend the result to the whole box V N .Their technique was later refined by Daviaud [15] who computed the log-number of high points in V δ N : for 0 < λ < 1, It is a simple exercise to show using the above results that the free energy in V N of the model is given by (1.6) Indeed, there is the clear lower bound log v∈V N e βφv ≥ log v∈V δ N e βφv , which can be evaluated using the log-number of high points (1.5) by Laplace's method.The upper bound is obtained using a comparison argument with i.i.d.centered Gaussians.
A striking fact is that the three above results correspond to the expressions for N 2 independent Gaussian variables of variance 1 π log N 2 .In other words, correlations have no effects on the above observables of the extremes.The purpose of the paper is to extend this correspondence to observables related to the Gibbs measure.
To this aim, consider the normalized Gibbs weights or Gibbs measure where Z N (β) := v∈V N e βφv .We consider the normalized covariance or overlap This is the covariance divided by the dominant term of the variance in the bulk.
In spin glasses, the relevant object to classify the extreme value statistics of strongly correlated variables is the two-overlap distribution function The main result shows that the 2D GFF falls within the class of models that exhibit a one-step replica symmetry breaking at low temperature.
Note that for β ≤ β c , it follows from (1.6) that the overlap is 0 almost surely.The result is the analogue for the 2D GFF of the results obtained by Derrida & Spohn [17] and Bovier & Kurkova [10,11] for the branching Brownian motion and for GREMtype models.In [4], such a result was proved for a non-hierarchical log-correlated Gaussian field constructed from the multifractal random measure of Bacry & Muzy [5], see also [22] for a closely related model.This type of result was conjectured by Carpentier & Ledoussal [14].We also remark that Theorem 1.1 shows that at low temperature two points sampled with the Gibbs measure have overlaps 0 or 1.This is consistent with the result of Ding & Zeitouni [19] who showed that the extremal values of GFF are at distance from each other of order one or of order N .
A general method to prove Poisson-Dirichlet statistics for the distribution of the overlaps from the one-step replica symmetry breaking was laid down in [4].This connection is done via the (now fundamental) Ghirlanda-Guerra identities.Another equivalent approach would be using stochastic stability as developed in [1,2,3].The reader is referred to Section 2.3 of [4] where the connection is explained in details for general Gaussian fields.For the sake of conciseness, we simply state the consequence for the 2D GFF.
Consider the product measure → R be a continuous function.Write F (q ll ) for the function evaluated at q ll := q(v l , v l ), l = l , for (v 1 , . . ., v s ) ∈ V ×s N .We write EG ×s β,N F (q ll ) for the averaged expectation.Recall that a Poisson-Dirichlet variable ξ of parameter α is a random variable on the space of decreasing weights s = (s 1 , s 2 , . . . ) with 1 ≥ s 1 ≥ s 2 ≥ • • • ≥ 0 and i s i ≤ 1 which has the same law as η i / j η j , i ∈ N ↓ where ↓ stands for the decreasing rearrangement and η = (η i , i ∈ N) are the atoms of a Poisson random measure on (0, ∞) of intensity measure s −α−1 ds.
The theorem below is a direct consequence of the Theorem 1.1, the differentiability of the free energy (1.6) as well as Corollary 2.5 and Theorem 2.6 of [4].→ R of the overlaps of s replicas: The above is one of the few rigorous results known on the Gibbs measure of logcorrelated fields at low temperature.Theorem 1.2 is a step closer to the conjecture of Duplantier, Rhodes, Sheffield & Vargas (see Conjecture 11 in [20] and Conjecture 6.3 in [30]) that the Gibbs measure, as a random probability measure on V N , should be atomic in the limit with the size of the atoms being Poisson-Dirichlet.Theorem 1.2 falls short of the full conjecture because only test-functions of the overlaps are considered.Finally, it is expected that the Poisson-Dirichlet statistics emerging here is related to the Poissonian statistics of the thinned extrema of the 2D GFF proved by Biskup & Louidor in [6] based on the convergence of the maximum established by Bramson, Ding & Zeitouni [12].To recover the Gibbs measure from the extremal process, some properties of the cluster of points near the maxima must be known.
The rest of this paper is dedicated to the proof of Theorem 1.1.In Section 2, a generalized version of the GFF (whose variance is scale-dependent) is introduced.It is a kind of non-hierarchical GREM and is related to a model studied by Fyodorov & Bouchaud in [23].The proof of Theorem 1.1 is given in Section 3. It relates the overlap distribution of the 2D GFF to the free energy of the generalized GFF.The free energy of the generalized GFF needed in the proof is computed in Section 4.

The multiscale decomposition and a generalized GFF
In this section, we construct a Gaussian field from the GFF whose variance is scaledependent.The construction uses a multiscale decomposition along each vertex.The construction is analogous to a Generalized Random Energy Model of Derrida [16], but where correlations are non-hierarchical.Here, only two different values of the variance will be needed though the construction can be directly generalized to any finite number of values.Consider 0 < t < 1.We assume to simplify the notation that N 1−t is an even integer and that N t divides N .The case of general t's can also be done by making trivial corrections along the construction.
For v ∈ V N , we write [v] t for the unique box with N 1−t points on each side and centered at v. If [v] t is not entirely contained in V N , we take the convention that [v] t is the intersection of the square box with V N .For t = 1, take where the second equality holds by the Markov property of the Gaussian free field, see Lemma 5.1.Clearly, for any v ∈ V N , the random variable φ [v]t is Gaussian.Moreover, by Lemma 5.1, is the probability that a simple random walk starting at v hits u at the first exit time of [v] t .
The following multiscale decomposition holds trivially The decomposition suggests the following scale-dependent perturbation of the field.For 0 < α < 1 and σ The Gaussian field (ψ v , v ∈ V N ) will be called the (α, σ)-GFF on V N .
To control the boundary effects, it is necessary to consider the field in a box slightly smaller than V N .For ρ ∈ (0, 1), let where The associated free energy is given by (Note that log #A N,ρ = (1 + o N (1)) log N 2 .)Its L 1 -limit is a central quantity needed to apply Bovier-Kurkova technique.This limit is better expressed in terms of the free energy of the REM model consisting of N 2 i.i.d.Gaussian variables of variance Theorem 2.1.Fix α ∈ (0, 1) and σ = (σ 1 , σ 2 ) ∈ R 2 + and let V 12 := σ 2 1 α + σ 2 2 (1 − α).Then, for any ρ < α, and for all β > 0 (2.6) where the convergence holds almost surely and in L 1 .
Note that the limit does not depend on ρ.
3. Proof of Theorem 1.1 3.1.The Gibbs measure close to the boundary.The first step in the proof of Theorem 1.1 is to show that points close to the boundary do not carry any weight in the Gibbs measure of the GFF in V N .The result would not necessarily hold if we considered instead the outside of V δ N which is much larger than the outside of A N,ρ .Lemma 3.1.For any ρ > 0, (3.1) lim Before turning to the proof, we claim that the lemma implies that, for any r ∈ [0, 1] and ρ ∈ (0, 1), is the two-overlap distribution of the Gibbs measure of the GFF (φ for Z N,ρ (β) := v∈A N,ρ e βφv .Indeed, introducing an auxiliary term ).The second term equals which is also smaller than 2 EG β,N (A c N,ρ ).Lemma 3.1 then implies (3.2) as claimed.
Proof of Lemma 3.1.Let > 0 and λ > 0. The probability can be split as follows where f (β) is defined in (1.6).The second term converges to zero by (1.6).The first term is smaller than Since the free energy is a Lipschitz function of the variables φ v , see e.g.Theorem 2.2.4 in [31], the free energy self-averages, that is for any t > 0 lim To conclude the proof, it remains to show that for some C < 1 (independent of N but dependent on ρ) Note that by Lemma 5.1, the maximal variance of N,ρ ) independent centered Gaussians (and independent of (φ v ) v∈A c N,ρ ) with variance given by E[g Moreover, by a standard comparison argument (see Lemma 5.3 in the Appendix), is smaller than the expectation for i.i.d.variables with identical variances.The two last observations imply that ), the free energy of these i.i.d.Gaussians in the limit N → ∞ is given by (2.5) The last two equations then imply lim sup It is then straightforward to check that, for every β, the right side is strictly smaller than f (β) as claimed.Proof.Without loss of generality, we suppose that lim ρ→0 lim N →∞ x β,N,ρ = x β in the sense of weak convergence.Uniqueness of the limit x β will then ensure the convergence for the whole sequence by compactness.Note also that by right-continuity and monotonicity of x β , it suffices to show (3.6) We can choose a dense set of α such that none of them are atoms of x β , that is Now recall Theorem 2.1.Pick σ = (1, 1 + u) for some parameter |u| ≤ 1.Since β > √ 2π, u can be taken small enough so that β is larger than the critical β's of the limit.The goal is to establish the following equality: The conclusion follows from this equality.Indeed, by construction, the function u → f (α,σ) N,ρ (β) is convex.In particular, the limit of the derivatives is the derivative of the limit at any point of differentiability.Therefore, a straightforward calculation from (2.6) with σ 1 = 1 and σ 2 = 1 + u gives: (3.8) lim This gives (3.6) at u = 0.
We introduce the notation for the overlap at scale α: The first identity holds since by Fubini's theorem The identity is then obtained by Gaussian integration by parts.
To prove (3.7), we need to relate the overlap at scale α with the overlap as well as the event {q(v, v ) ≥ α} with the event {v ∈ [v] α }.This is slightly complicated by the boundary effect present in GFF.The equality in the limit N → ∞ between the first terms of (3.10) and (3.11) .

Therefore, we have for
It remains to establish the equality between the second terms of (3.10) and (3.11).
Here, a control of the boundary effect is necessary.The following observation is useful to relate the overlaps and the distances: if v, v ∈ A N,ρ , Lemma 5.2 gives On one hand, the right inequality proves the following implication for some constant c independent of N and ρ.On the other hand, the left inequality gives: Using this, we show To establish the equality of the overlaps on the event {q(v, v ) ≥ α + ε}, consider the decomposition, . Therefore, the first term of the right side of (3.17) is by Lemma 5.2 .
The second term is negligible.Indeed, by Cauchy-Schwarz inequality, it suffices to prove that For this, write B for the box [v] α ∩ [v ] α .We have ) by Lemma 5.2 and the fact that distances of v to vertices in ∂ B and Equations (3.12), (3.16) and (3.20) yield ∆ 1 (N, ρ) → 0 in the limit N → ∞, ρ → 0 and ε → 0.
4. The free energy of the (α, σ)-GFF: proof of Theorem 2.1 The computation of the free energy of the (α, σ)-GFF is divided in two steps.First, an upper bound is found by comparing the field ψ in A N,ρ with a "non-homogeneous" GREM having the same free energy as a standard 2-level GREM.Second, we get a matching lower bound using the trivial inequality f The limit of the right term is computed following the method of Daviaud [15].4.1.Proof of the upper bound.For conciseness, we only prove the case σ 1 ≥ σ 2 , by a comparison argument with a 2-level GREM.The case σ 1 ≤ σ 2 is done similarly by comparing with a REM.The comparison argument will have to be done in two steps to account for boundary effects.
Divide the set A N,ρ into square boxes of side-length N 1−α /100.(The factor 1/100 is a choice.We simply need these boxes to be smaller than the neighborhoods [v] α , yet of the same order of length in N .)Pick the boxes in such a way that each v ∈ A N,ρ belongs to one and only one of these boxes.The collection of boxes is denoted by B α and ∂B α denotes B∈Bα ∂B.For v ∈ A N,ρ , we write B(v) for the box of B α to which v belongs.For B ∈ B α , denote by B ⊃ B the square box given by the intersections of all [u] α , u ∈ B, see figure 1. Remark that the side-length of B is cN 1−α , for some constant c.For short, write φ The idea in constructing the GREM is to associate to each point v ∈ B the same contribution at scale α, namely φ B .One problem is that φ B will not have the same variance for every B since it depends on the distance to the boundary.This is the reason why the comparison will need to be done in two steps.First, consider the hierarchical Gaussian field ( ψ v , v ∈ A N,ρ ): (4.1) B(v) + g (2)  v , where (g v , v ∈ A N,ρ ) are independent centered Gaussians (also independent from (g B , B ∈ B α ) are also independent centered Gaussians with variance chosen to be σ 2 1 E[φ 2 B ] + C for some constant C ∈ R independent of B in B α and independent of N .The next lemma ensures that Moreover, if v and v both belong to B ∈ B α , then For the first assertion, write u) φ u of Lemma 5.1 and the fact that σ 1 > σ 2 imply that E[ψ v ψ v ] ≥ 0 since the field φ is positively correlated by (1.1).
Suppose now that v, v ∈ B where B ∈ B α .The covariance can be written as We first prove that the last two terms of (4.3) are positive.By Lemma 5.1, we can write φ Lemma 5.2 ensures that the correlation in the sum are positive.
For the first term of (4.3), the idea is to show that φ[v] α and φ B are close in the L 2 -sense.The same argument used to prove (3.19) shows that Moreover, since v and v B are also at a distance smaller than N 1−α /100 from each other, Lemma 12 in [7] implies that and similarly for v .All the above sum up to . It remains to show that the second term of (4.3) is greater than O N (1).Since φ [v]α and φ [v ]α are F B c -measurable by definition of the box B, we have the decomposition where Z N (β) = v e βXv .Gaussian integration by part then yields The right side is clearly positive, hence proving the lemma.
4.2.Proof of the lower bound.Recall the definition of V δ N given in the introduction.The two following propositions are used to compute the log-number of high points of the field ψ in V δ N .The treatment follows the treatment of Daviaud [15] for the standard GFF.The lower bound for the free energy is then computed using Laplace's method.Define for simplicity where Proof.The case σ 1 ≤ σ 2 is direct by a union bound.In the case σ 1 ≥ σ 2 , note that the field ψ defined in (4.1) but restricted to V δ N is a 2-level GREM with cN 2α (for some c > 0) Gaussian variables of variance at the first level.Indeed, for the field restricted to V δ N , the variance of E[φ 2 B ] is 1) by Lemma 5.2 since the distance to the boundary is a constant times N .Therefore, by Lemma 5.3 and Equation (4.2), we have The result then follows from the maximal displacement of the 2-level GREM.We refer the reader to Theorem 1.1 in [10] for the details.

appendix
The conditional expectation of the GFF has nice features such as the Markov property, see e.g.Theorems 1.2.1 and 1.2.2 in [21] for a general statement on Markov fields constructed from symmetric Markov processes.The following estimate on the Green function can be found as Lemma 2.2 in [18] and is a combination of Proposition 4.6.2 and Theorem 4.4.4 in [28].
Lemma 5.2.There exists a function a : (where γ 0 denotes the Euler's constant) such that a(v, v) = 0 and Slepian's comparison lemma can be found in [27] and in [26] for the result on logpartition function.

1 . Introduction 1 . 1 .
The model.Consider a finite box A of Z 2 .The Gaussian free field (GFF) on A with Dirichlet boundary condition is the centered Gaussian field (φ v , v ∈ A) with the covariance matrix(1.1)

Figure 1 .
Figure 1.The box B ∈ B α the corresponding box B which is the intersection of all the neighborhoods [v] α , v ∈ B.

Lemma 5 . 1 .
Let B ⊂ A be subsets of Z 2 .Let (φ v , v ∈ A) be a GFF on A. Then E[φ v |F B c ] = E[φ v |F ∂B ], ∀v ∈ B,and(φ v − E[φ v |F ∂B ], v ∈ B)has the law of a GFF on B.Moreover, if P v is the law of a simple random walk starting at v and τ B is the first exit time of B, we haveE[φ v |F ∂B ] = u∈∂B P v (S τ B = u) φ u .