Fluctuations of maxima of discrete Gaussian free fields on a class of recurrent graphs ∗

We provide conditions that ensure that the maximum of the Gaussian free field on a sequence of graphs fluctuates at the same order as the field at the point of maximal standard deviation; under these conditions, the expectation of the maximum is of the same order as the maximal standard deviation. In particular, on a sequence of such graphs the recentered maximum is not tight, similarly to the situation in Z but in contrast with the situation in Z. We show that our conditions cover a large class of “fractal” graphs.


Introduction
The study of the maxima of Gaussian fields has a rich history, which we will not attempt to survey here.The general theory was developed in the 70s and 80s, and an excellent account can be found in [19].However, general results concerning the order of fluctuations of the maximum are lacking, except for some fundamental inequalities such as the Borell-Tsirelson inequality and the recent work of Chatterjee [9] showing the equivalence of superconcentration (where the fluctuations of the maximum are of a lower order than the maximal standard deviation of the field) to the chaos property (where the location of the maximum is sensitive to small perturbations of the field).
In recent years, a special effort has been directed toward the study of the so called Gaussian free field (GFF) on various graphs.While we postpone the general definition to the next section, we discuss in this introduction the special case of the GFF on subsets V N = ([−N, N ] ∩ Z) d , with Dirichlet boundary conditions.These are random fields {X x } x∈V N indexed by points in V N , with joint density (with respect to Lebesgue measure) proportional to exp −c x∼y (X x − X y ) 2 , with the sum over neighbors in V N , and X x = 0 for x ∈ ∂V N .(An alternative description involving the Green function of random walk on V N is given below in Section 2; see also [21] for a very readable introduction to GFFs in a continuous setting.)With X * N,d denoting the maximum of the GFF on V N in dimension d, it is not hard to see that X * N,d is of order √ N for d = 1, order log N for d = 2, and order (log N ) 1/2 for d ≥ 3.Moreover, a consequence of the Borell-Tsirelson inequality (see [19]) is that for d ≥ 3, since simple random walk is transient on Z d , the fluctuations of X * N,d are at most of order 1, while for d = 1 the fluctuations of X * N,1 are of the same order as X * N,1 , i.e. of order

√
N .The critical case d = 2 was settled only recently [8], where it was shown that the fluctuations of X * N,2 are also of order 1.This raises naturally the question of determining for which sequences of graphs is the sequence of recentered maxima of the GFF tight.Our goal in this paper is to exhibit a class of sequences of graphs, which are fractal-like and for which the maximum of the GFF fluctuates at the same order as the maximum itself, and both are of the order of the maximal standard deviation of the GFF in the graph.In that respect, the behavior of the maximum is similar to that of X * N,1 .For this class of graphs, we also show that the cover time of the graph, measured in terms of the (square root of the) local time at a fixed vertex, also does not concentrate.(We note in passing that for V N in two dimensions, it is, to the best of our knowledge, an open problem to decide whether this quantity concentrates or not.) The structure of the paper is as follows.In the next section, we introduce the GFF on general graphs and state Assumption 2.1 that characterizes the graphs which we investigate; the main feature is a relation between the graph distance and the resistance, and control of the covering number of the graph in terms of resistance distance.We then state our main result, Theorem 2.2, concerning fluctuations of the maximum of the GFF.We also state Proposition 2.4 concerning the cover time of the graphs.Proofs of the theorem and proposition are given in Section 3. The heart of the paper is then Section 4.1, where we show that certain naturally constructed fractal-like graphs satisfy our assumptions.In particular, this is the case for the standard Sierpinski carpets in two dimensions and gaskets in all dimensions.
Notation Throughout the paper, we use c 1 , c 2 , • • • to denote generic constants, independent of N , whose exact values are not important and may change from line to line.We write a n

Framework
We first introduce general notation for finite graphs with a 'wired' boundary and their associated resistance.Let G = (V (G), E(G)) be a connected (undirected) finite graph with at least two vertices, where V (G) denotes the vertex set and E(G) the edge set of G. Let d G be the graph distance, that is, d G (x, y) is the number of edges in the shortest path from x to y in G. Define a symmetric weight function µ G : V (G) × V (G) → R + that satisfies µ G xy > 0 if and only if {x, y} ∈ E(G).For B ⊂ G with B = G and for distinct x, y ∈ V (G) not both in B, we define the resistance between x and y by for any y ∈ B. We write R(x, y) := R ∅ (x, y).
The resistance R B (•, •) is the resistance of the following electrical network with a 'wired' boundary: Consider the graph G obtained by combining all vertices in B to a single vertex b, that is Define the modified symmetric weight function and set as before µ G x = y∈V (G) µ G xy .Let {w t } t≥0 be the continuous time random walk on G such that the holding time at a vertex is exp(1), and the jump probability is given by µ x,y /µ x .Let and the corresponding continuous time Markov chain {w N t } t≥0 with the wired boundary condition on B N as above.We assume that G N \ B N is connected.Let T N := min{t ≥ 0 : w N t = b}, and define, for each x, y ∈ G N denotes the expectation with respect to w N t started at x.For z ∈ B N , we set X N z ≡ 0. The Gaussian free field (GFF for short) on G N (with boundary B N ) is the zero-mean Gaussian field It can be easily checked (using for instance [10, Lemma 2.1], [16,Proposition 3.6]) that y).Let h : N → N be a strictly increasing function with h(0) = 0, that satisfies the following doubling property: there exist 0 We assume the following.

Assumption 2.1.
There exist α > 0 and c 1 , c 2 , c 3 > 0 such that the following hold for all large N .

P ( XN
In particular, under Assumption 2.1, {X * N − EX * N } N fluctuates with order σ N and therefore it is not tight.
Remark 2.3.We stated Assumption 2.1 with respect to the graph distance in G N , because this will be easiest to check in the applications.However, one should note that the proof of Theorem 2.1 does not depend on the particular metric chosen, as long as the metric satisfies the assumption.In particular, if we choose R B N (•, •) as the metric, Assumption 2.1 (i), (ii) turns out to be trivial with h(s) = s, and the assumption boils down to N R B N (δσ 2 N ) ≤ c 3 δ −α for all δ ∈ (0, 1] and lim In a recent seminal work, [10] have established a close relation between the expectation of the maximum of the GFF on general graphs and the expected cover time of these graphs by random walk.Under the assumptions of Theorem 2.2, one can also derive information on the fluctuations of the cover time, as follows.Define the cover time of G N as It is easy to see We will consider the square-root of the normalized local time at B N at cover time, i.e. the random variable L N := L b,N τ N cov .One expects (see [10]) that L N should behave similarly to |X * N |.In the special case of G N being the rooted at b binary tree of depth N , this was confirmed in [7].In our setup here, this is confirmed in the following proposition.

Proofs of Theorem 2.2 and Proposition 2.4
We begin with the proof of Theorem 2.2.
Proof of Theorem 2.2: Then, using As- sumption 2.1 (i),(ii), there exists c > 0 such that for all x, y ∈ G N with d G N (x, y) ≤ d N max and all Thus, denoting N d(ε) the minimal number of d-balls of radius ε needed to cover G N , we have where we used Assumption 2.1 (iii) in the second inequality.Rewriting this, we have , where c > 0 is independent of N .Set γ = 2α/β 1 .We can apply standard metric entropy bounds (for this version, see [1, Theorem 5.2]) to deduce that there exist λ 0 > 0 and N 0 such that for all λ > λ 0 , ε > 0 and N > N 0 , where C γ ≥ 1 does not depend on N and Ψ(λ) = (2π) −1/2 ∞ λ e −x 2 /2 dx.On the other hand, let x * N be such that E(X 2 Then, for any λ > 0, The estimates in (2.2) are easy consequences of the last two displayed inequalities.

2
We turn to the analysis of cover times.Proof of Proposition 2.4: The upper bound in the proposition is a consequence of the Eisenbaum-Kaspi-Marcus-Rosen-Shi isomorphism theorem [11], as was observed in [10]: indeed, by [10, Eq. ( 20), (21)] and using the last estimate in (2.2), there exist constants c 1 , c 2 > 0 so that with t = θσ 2 N , and all θ large enough, P (min while where τ N (t) := inf{s > 0 : L b,N s > t}.On the event {min x L x,N τ N (t) ≥ t/2} we have that τ N (t) ≥ τ N cov .Thus, on the event In particular, (3.1), (3.2) and (3.3) imply that EL N /σ N is bounded uniformly.
To estimate L N from below, we use the Markov property.Let We decompose the walk w N t according to excursions from b: the probability to hit x * during one excursion (see e.g. [20, Ch. 2]) is where Z N is geometric of parameter p N and E i are standard independent exponential random variables.Note that EL b,N T x * = σ 2 N .
Consider now a parameter ξ > 0. We have that Note that from the properties of the geometric distribution, regardless of p N we have that , and in any case we also have that P 2 ≥ c 2 (ξ) > 0. We conclude that

Nested fractal graphs and strongly recurrent Sierpinski carpet graphs
where U i is a unitary map and γ i ∈ R d .We assume that {ψ i } K i=1 satisfies the open set condition, namely there exists a non-empty bounded set i=1 is a family of contraction maps, there exists a unique non-empty compact set F such that F = ∪ K i=1 ψ i (F ).We assume that F is connected.V 0 := {x ∈ Ξ : ∃i, j ∈ {1, . . ., K}, i = j and y ∈ Ξ such that ψ i (x) = ψ j (y)} .
Assume that #V 0 ≥ 2 and set ψ i1...in := F is then called a nested fractal if the following holds.
• (Symmetry) If x, y ∈ V 0 , then the reflection in the hyperplane We assume without loss of generality that ψ 1 (x) = L −1 x and that the origin belongs to V 0 .Let V (G N ).Next, define B 0 := {{x, y} : we place a copy of B 0 and denote by B the set of all the edges determined in this way.Next, we assign µ xy = µ yx > 0 for each {x, y} ∈ B in such a way that there exist c 1 , c 2 > 0 such that We call the graph (G, µ) a nested fractal graph.A typical example is the 2-dimensional Sierpinski gasket graph in Fig 1 (where L = 2).Let d(•, •) be the graph distance on G, {w k } k the Markov chain for (X, µ), and define the heat kernel as p k (x, y) = P x (w k = y)/µ y .(Note that we consider the discrete time Markov chain here in order to apply the results in [5] to derive the resistance estimates (4.5).Indeed, (4.5) can be obtained through both discrete and continuous time Markov chains.)It is known (see [13] (also [17] for the continuous setting)) that there exist constants c 3 , . . ., c 6 such that for all x, y ∈ G, k > 0 We now define a sequence of graphs {G N } N ≥0 by setting V (G N ) as above and E(G . So (4.5) implies Assumption 2.1 (i),(ii) with h(s) = s dw−d f , and (4.4) with the self-similarity of the graph imply Assumption 2.1 (iii) with α = d f .We note that we can actually take B N arbitrary as long as d N max (Lη) N .

Strongly recurrent Sierpinski carpet graphs
i=1 be a family of L-similitudes of H 0 onto some element of Q.We assume that the sets ψ i (H 0 ) are distinct, and as before assume ψ 1 (x) = L −1 x.Set H 1 = ∪ K i=1 ψ i (H 0 ).Then, there exists a unique non-void compact set F ⊂ H 0 such that F = ∪ K i=1 ψ i (F ).We assume F is connected.F is called a (generalized) Sierpinski carpet if the following hold (cf.[4]): (SC1) (Symmetry) H 1 is preserved by all the isometries of the unit cube H 0 .(SC2) (Non-diagonality) Let B be a cube in H 0 which is the union of The main difference from nested fractals is that Sierpinski carpets are infinitely ramified, i.e.F cannot be disconnected by removing a finite number of points.
Let V 0 be a set of vertices in H 0 and define V (G N ) and G as in (4.1).Set B 0 := {{x, y} : x = y ∈ V 0 , |x − y| = 1}, and define B and µ xy as in the case of nested fractal graphs.We call the graph (G, µ) a Sierpinski carpet graph.A typical example is the 2-dimensional Sierpinski carpet graph in It is known, see [3] and also [4] for the continuous setting, that (4.2), (4.3) hold, where d w = log(ρK)/ log L, d f = log K/ log L with some constant ρ > 0. For the 2-dimensional Sierpinski gasket graph, L = 3, K = 8 and ρ > 1.Let us restrict ourselves to the case ρ > 1, namely d w > d f .In this case, since (4.4) holds, we can show that (4.2) and (4.3) imply (4.5) as before.Arguing further as before, we have Assumption 2.1 (i)-(iii) with h(s) = s dw−d f and α = d f .

Homogeneous random Sierpinski carpet graphs
Let ≥ 2 and I := {1, • • • , }.For each k ∈ I, let {ψ k i } K k i=1 be a family of L k -similitudes as in the definition of the Sierpinski carpet graphs.As before, we assume ψ k V (G N ξ| N ).
For x ∈ G ξ and r ≥ 1, let V d (x, r) be the number of vertices in the ball of radius r centered at x w.r.t.
the graph distance.It can be easily seen that We set τ (0) = h(0) = 0. Note that τ and h satisfy the property in (2.1) since < ∞.
Given these, it is possible to obtain heat kernel estimates similar to those in Theorem 6.3 and Lemma 6.7 of [14] by tracking the proof in [14] faithfully (see the Appendix for a sketch).By making additional computations (similar to those in [12,Lemma 3.19]) in the proof of [14, Lemma 3.10], we can obtain the following heat kernel estimates (cf.Remark after Theorem 24.6 in [15]): There exist for k ≥ c 6 τ (d(x, y)). (4.9) Now assume the following limits exist and the inequality holds.
Under this assumption, we have , ∀x, y ∈ G ξ . (4.11) The equivalence of (4.8)+(4.9)and (4.11) is proved in [5] when τ (s) = s β for some β ≥ 2 under some volume growth condition referred as (V G(β − )).It is an interesting open problem to prove whether the recentered maximum of the GFF is tight or not when p = p * .Note that the method in [8] cannot be directly applied since it relies on detailed comparisons with a translation invariant graph.

A Appendix: Heat kernel estimates for Markov chains on homogeneous random Sierpinski carpet graphs
In this appendix, we will briefly sketch the proof of (4.8) and (4.9).The Markov chain we consider here is the discrete time Markov chain.

Proposition 2 . 4 .
With notation as above and under Assumption 2.1, the conclusion of Theorem 2.2 hold with L N /σ N replacing XN .

(4. 7 )
Define a time scale function τ : [1, ∞) → [1, ∞) and resistance scale factor y) for x, y ∈ G N ξ| N and d N max B N .So (4.11) implies Assumption 2.1 (i),(ii), and (4.7), (4.10) with the homogeneity of the graph imply Assumption 2.1 (iii) with α = max n d f (n).As before we can take B ξ| N arbitrary as long as d N max B N .Finally we will introduce randomness on this graph.Let (I N , F, P) be a Borel probability space where the measure P is stationary and ergodic for the shift operator θ :I N → I N defined by θ((k 1 , • • • , k n , • • • )) = (k 2 , • • • , k n , • • • ).Then, by [14, Proposition 7.1] and the sub-additive ergodic theorem, one can prove the existence of the first two limits in (4.10).Let d i f , d i w be the Hausdorff dimension and the walk dimension for G i where i = (i, i, i, • • • ) for i ∈ I. Let us consider a special case when d = 3, = 2, and P is the Bernoulli probability measure with P(ξ 1 = 1) = p, P(ξ 1 = 2) = 1−p for some p ∈ [0, 1].
Now, let {G N } N ≥1 be a sequence of finite connected graphs such that |G N | ≥ 2 for all N ≥ 1 and lim [5,e we need a generalized version of this under the doubling property of τ .In fact, we only need (4.8)+(4.9)⇒(4.11),and the generalization of this direction is easy.Indeed, using (4.8) and (4.9), we can obtain the scaled Poincaré inequality and the lower bound of (4.11) similarly to the proof of [5, Proposition 4.2] (with τ (s) replacing s β there).Under (4.10), a condition corresponding to (V G((d w ) − )) in[5]holds, so together with the scaled Poincaré inequality, we can obtain the upper bound of (4.11) similarly to the proof of[5, Lemma 2.3  (b)].
Now let B ξ| N One can see that d f /d w is a continuous function of p. Indeed, it can be easily seen that it is enough to prove lim n→∞ log R n /n is continuous for p.By the proof of [14, Proposition 7.1], there exist c 1 , c 2 > 0 such that we have1 k E log(c 1 R k ) ≤ lim R k ), P − a.s.,for any k ≥ 1 where E is the average over P. Since E log(c i R k ), i = 1, 2 are continuous for p (because the graph is finite), we obtain the desired continuity of lim n→∞ log R n /n.So, when we choose the two carpets in such a way that d 1 w > d 1 f and d 2 w < d 2 f (which is possible, see [4, Section 9]), we are able to construct a one parameter family of homogeneous random Sierpinski carpet graphs where d f /d w is P-a.e. an arbitrary fixed number between d 1 f /d 1 w and d 2 f /d 2 w .In particular, there exists p * ∈ (0, 1) such that (4.10) holds P-a.e. for all p < p * .