Recursions and tightness for the maximum of the discrete, two dimensional Gaussian Free Field

We consider the maximum of the discrete two dimensional Gaussian free field in a box, and prove the existence of a (dense) deterministic subsequence along which the maximum, centered at its mean, is tight; this still leaves open the conjecture that tightness holds without the need for subsequences. The method of proof relies on an argument developed by Dekking and Host for branching random walks with bounded increments and on comparison results specific to Gaussian fields.


Introduction and main result
We consider the discrete Gaussian Free Field (GFF) in a two-dimensional box of side N + 1, with Dirichlet boundary conditions. That is, let V N = ([0, N ] ∩ Z) 2 , V o N = ((0, N )∩Z) 2 , and let {w m } m≥0 denote a simple random walk started in V N and killed at τ = min{m : w m ∈ ∂V N } (that is, killed upon hitting the boundary . For x, y ∈ V N , define G N (x, y) = E x ( τ m=0 1 wm=y ), where E x denotes expectation with respect to the random walk started at x. The GFF is the zero-mean Gaussian field {X N z } z indexed by z ∈ V N with covariance G N . Let X * N = max z∈VN X N z . It was proved in [5] that X * N /(log N ) → c with c = 2 2/π, and the proof is closely related to the proof of the law of large numbers for the maximal displacement of a branching random walk (in R). Based on the analogy with the maximum of independent Gaussian variables and the case of branching random walks, the following is a natural conjecture.

Conjecture 1 The sequence of random variables
To the best of our knowledge, the sharpest result in this direction is due to [7], who shows that the variance of Y N is o(log N ); in the same paper, Chatterjee also analyzes related Gaussian fields, but in all these examples, does not prove tightness. We defer to Section 3 for some pointers to the relevant literature concerning the Gaussian free field and the origin of Conjecture 1. The goal of this note is to prove a weak form of the conjecture. Namely, we will prove the following.
More information on the sequence {N k } k≥1 is provided below in Remark 1.
It is of course natural to try to improve the tightness from subsequences to the full sequence. As will be clear from the proof, for that it is enough to prove the existence of a constant C such that EX * 2N ≤ EX * N + C. This is weaker than, and implied by, the conjectured behavior of EX * N , which is for c = 2 2/π and an appropriate c 2 , see e.g. [6] and Remark 3. Finally, although we deal here exclusively with the GFF, it should be clear from the proof that the analysis applies to a much wider class of models.

Preliminary considerations
Our approach is motivated by the proof of tightness of branching random walks (BRW) with independent increments, in the spirit of [9] (see also the argument in [3]). We will thus first introduce a branching-like structure in the GFF. Unfortunately, this structure is not directly suitable for analysis, and so we later modify it.

The basic branching structure
We consider N = 2 n in what follows, write Z n = X * N and identify an integer m = n−1 ℓ=0 m i+1 2 i with its binary expansion (m n , m n−1 , . . . , m 1 ). For k ≥ 1, introduce the sets of k-diadic integers x i , y i denoting as above the ith digit in the binary expansion of x, y. We introduce the random variables We then have the decomposition where, by the Markov property of the GFF, the collections {X z1 · } z1∈V1 are i.i.d. copies of the GFF in the box V N/2 , and are independent of the collection of random variables {ξ z1 · }. Iterating, we have the representation where all the summands in the right side of (4) are independent. Recall that We can now explain the relation with branching random walks: should the random variables in the right side of (4) not depend on the conditioning (that is, the superscript), (4) would correspond precisely to a branching random walk (on a four-ary tree), with time-dependent increments. For such BRW, a functional recursion for the law of X * N can be written down, and used to prove tightness (see [4] and [1]). Unfortunately, no such simple functional recursions are available in the case (4). For this reason, we first modify the representation (4), and then adapt an argument of [9], originally presented in the context of BRW. To explain our goal, note that we have from (3) that where the variables {(X * N/2 ) z } z are four i.i.d. copies of X * N/2 , and the D z,N are complicated fields but D z,N z2,...,zn ≥ min z2,...,zn ξ z z2,...,zn . Unfortunately, the D z,N variables are far from being uniformly bounded, and this fact prevents the application of the argument from [9].

Two basic lemmas
In this subsection we present two preliminary lemmas that will allow for a comparison of the GFF between different scales. The first shows that the maximum of the sum of two zero mean fields tends to be larger than each of the fields.
Lemma 1 Let {X i } i∈VN i and {Y i } i∈VN be two independent random fields and assume that EY i = 0. Then, Proof Let α ∈ V N be such that max i∈VN X i = X α (in case several αs satisfy the above equality, choose the first according to lexicographic order). We then have where the second equality is due to the independence of the fields and the fact that EY i = 0 for all i.

By (5) and Lemma 1, we have that
The following lemma gives a control in the opposite direction.
Lemma 2 There exists a sequence n k → ∞ and a constant C such that Proof From [5] there exists a constant c > 0 so that EZ n /n → c. Fixing arbitrary K and defining I n,K = {i ∈ [n, 2n] : EZ n+1 > EZ n + K}, one has from (7) and the existence of the limit EZ n /n → c that In particular, choosing K = 3c it follows that for all n large, there exists an n ′ ∈ [n, 2n] so that EZ n ′ ≤ EZ n ′ −1 + K , as claimed.

Proof of Theorem 1
By (5) and Lemma 1, we get that whereX * N is an independent copy of X * N . Using the equality max(a, b) = (a + b + |a − b|)/2, we get that For the sequence n k and the constant C from Lemma 2, we thus get that This shows that the sequence {X * 2 n k } k≥1 is tight and completes the proof of Theorem 1.

Remark 1
The subsequence n k provided in Lemma 2 can be taken with density arbitrary close to 1, as can be seen from the following modification of the proof. Fixing arbitrary K and ǫ and defining I n,ǫ,K = {i ∈ [n, n(1 + ǫ)] : EZ n+1 > EZ n + K}, one has from (7) and the existence of the limit EZ n /n → c (with c = 2 2/π) that It is of course of interest to see whether one can take n k = k. Minor modifications of the proof of Theorem 1 would then yield Conjecture 1.

Some bibliographical remarks
The Gaussian free field has been extensively studied in recent years, in both its continuous and discrete forms. For an accessible review, we refer to [13]. The fact that the GFF has a logarithmic decay of correlation invites a comparison with branching random walks, and through this analogy a form of Conjecture 1 is implicit in [6]. This conjecture is certainly "folklore", see e.g. open problem #4 in [7]. For some one-dimensional models (with logarithmic decay of correlation) where the structure of the maxima can be analysed, we refer to [10,11]. The analogy with branching random walks has been reinforced by the study of the so called thick points of the GFF, both in the discrete form [8] and in the continuous form [12].