Thick points of random walk and the Gaussian free field

We consider the thick points of random walk, i.e. points where the local time is a fraction of the maximum. In two dimensions, we answer a question of [DPRZ01] and compute the number of thick points of planar random walk, assuming that the increments are symmetric and have a finite moment of order two. The proof provides a streamlined argument based on the connection to the Gaussian free field and works in a very general setting including isoradial graphs. In higher dimensions, we study the scaling limit of the thick points. In particular, we show that the rescaled number of thick points converges to a nondegenerate random variable and that the centered maximum of the local times converges to a randomly shifted Gumbel distribution.

V N = {−N, . . . , N } d , we denote τ N the first exit time of V N and ℓ t x , x ∈ Z d , t ≥ 0 the local times: In 1960, Erdős and Taylor [ET60] studied the behaviour of the local time of the most frequently visited site. By translating their work in our context of continuous time random walk, they proved that and conjectured that the limit also exists in dimension two and is equal to the upper bound. This conjecture was proved forty years later in a landmark paper [DPRZ01]. Estimates on the number of thick points, which are the points where the local times are larger than a fraction of the maximum, are also given in this paper. Briefly, their proof establishes the analogous results for the thick points of occupation measure of planar Brownian motion; taking in particular advantages of symmetries such as rotational invariance and certain exact computations on Brownian excursions. The discrete case is then deduced from the Brownian case through strong coupling/KMT arguments. This method requires all the moments of the increments to be bounded but the authors suspected that only finite second moments are needed. Later, the article [Ros05] showed that the paper [DPRZ01] can be entirely rewritten in terms of random walk giving a proof without using Brownian motion. This paper has two purposes. Firstly, we exploit the links between the local times and the Gaussian free field (GFF) provided by Dynkin-type isomorphisms to give a simpler and more robust proof of the two-dimensional result. The proof works in a very general setting (Theorem 3.1.1). In particular, we answer the question of [DPRZ01] about walks with only finite second moments and we also treat the case of random walks on isoradial graphs. Secondly, we obtain more precise results in dimension d ≥ 3. Namely, we show that the field {ℓ τN x , x ∈ V N } behaves like the field composed of i.i.d. exponential variables with mean E 0 [ℓ ∞ 0 ] located at each visited site by the walk. In particular, we show that the centered supremum of the local times as well as the rescaled number of thick points converge to nondegenerate random variables.
We first state two results on the planar case. Both are in fact corollaries of a more general theorem (Theorem 3.1.1) which will be stated later. We will then present the result in dimension d ≥ 3.

Dimension two
Consider Y t = S Nt , t ≥ 0, a continuous time random walk on Z 2 starting from the origin where S n = n i=1 X i , n ≥ 0, is the jump process with i.i.d. increments X i ∈ Z 2 and (N t ) t≥0 is an independent Poisson process of parameter 1. As before, we consider the square V N of side length 2N + 1, the first exit time τ N of V N and the local times ℓ t x , x ∈ Z 2 , t ≥ 0 defined as in (1). The theorem below shows that a = 1 corresponds to the maximum, and for 0 ≤ a ≤ 1, we call M N (a) the set of a-thick points Then we have the following: This theorem answers a question asked in the last section of [DPRZ01] with the additional assumption of symmetry. The assumption of symmetry is needed in our approach since otherwise we cannot define an associated GFF.
Our approach is sufficiently general that it can handle random walks with a very different flavour; for instance we discuss here the case of random walk on isoradial graphs.
We recall briefly the definitions and introduce some notations (we use the same as [CS11]). Let Γ = (V, E) be any connected infinite isoradial graph, with common radius 1, i.e. Γ is embedded in C and each face is inscribed into a circle of radius 1. Note that if x, y ∈ V are adjacent then x and y, together with the centers of the two faces adjacent to the edge {x, y}, form a rhombus. We denote by 2θ x,y the angle at x (or at y). See the figure 1 for an example. For instance, the square (resp. triangular, hexagonal, etc) lattice is an isoradial graph with θ x,y = π/4 (resp. π/6, π/3, etc) for all x ∼ y. We assume the following elliptic condition: , ∀x ∼ y, θ x,y ∈ η, π 2 − η .
Define ∀x ∼ y ∈ V the conductance c x,y = tan(θ x,y ) and let (Y t ) t≥0 be a Markov jump process with conductances (c e ) e∈E . Y is a continuous time walk which waits an exponential with mean 1/ y∼x c x,y time in each vertex x and then jumps from x to y with probability c x,y / z∼x c x,z . Take a starting point x 0 ∈ V and denoting d Γ the graph distance we define for all N ∈ N, and as before (equation (1)), we consider the first exit time τ N of V N and the local times. We will denote P x the law of the walk (Y t ) t≥0 starting from x ∈ V and E x the associated expectation.
As confirmed by the theorem below, a sensible definition of a-thick points is given by Theorem 1.1.2. We have the following two P x0 -a.s. limits: Remark 1.1.1. Theorems 1.1.1 and 1.1.2 also hold when we consider the walk up until a deterministic time, N 2 say, rather than the first exit time τ N of V N , since lim N →∞ log τ N / log N = 2 a.s. (easy to check but can also be seen from these two theorems). They also hold if we consider discrete time random walks rather than continuous time random walks. In that case, we have to multiply the discrete local times by the average time the continuous time walk stays in a given vertex before its first jump. See Remark 1.2.1 ending Section 1.2 for a small discussion about this.
Let us just confirm that Theorems 1.1.1 and 1.1.2 are coherent: in the square lattice case, the average time between successive jumps by the walk Y of Theorem 1.1.2 is 1/4 rather than 1.
It is plausible that the arguments of [Ros05] can be adapted to show Theorem 1.1.2. However, we include it here since it is a straightforward consequence of our approach (Theorem 3.1.1).

Higher dimensions
We now come back to the setting of the beginning of Section 1 for d ≥ 3 and we denote g := E 0 [ℓ ∞ 0 ]. In this section, the walk starts at the origin of Z d . We describe thick points through a more precise encoding by considering for a ∈ [0, 1] the point measure: Let us emphasize that the normalisation factor is equal to 1 when a = 1. We view ν a N as a random measure on [−1, 1] d × R. We compare the thick points of random walk with the thick points of i.i.d. exponential random variables with mean g located at each visited site by the walk. More precisely, we denote M N (0) := {x ∈ V N : ℓ τN x > 0} and taking E x , x ∈ Z d , i.i.d. exponential variables with mean g independent of M N (0), we define We finally denote by τ the first exit time of [−1, 1] d of Brownian motion starting at the origin and by µ occ the occupation measure of Brownian motion starting at the origin and killed at τ . Then we have: Moreover, for all a ∈ [0, 1) the distribution of ν a does not depend on a and ν a (dx, dℓ) (4) At criticality, ν 1 is a Poisson point process: We will see that this statement will imply the following two theorems: Theorem 1.2.2. If we define for every a ∈ [0, 1] the set of a-thick points: then there exist random variables M a such that for all a ∈ [0, 1] Moreover, for all a ∈ [0, 1) the distribution of M a does not depend on a and M a M 1 is a Poisson variable with parameter τ /g: for all k ≥ 0 Theorem 1.2.3. There exists an almost surely finite random variable L such that Moreover, L is a Gumbel variable with mode g log(τ /g) (location of the maximum) and scale parameter g, i.e. for all t ∈ R To the best of our knowledge, this result is not present in the current literature. A detailed study of the local times of random walk in dimension greater than two has been done in a series of papers by Csáki, Földes, Révész, Rosen and Shi (see [CFR07b] for a survey of this work). In particular, Theorem 1 of [Rév04] and the corollary following the main theorem of [CFR06] improved the estimate of Erdős and Taylor (equation (2)). By translating their work in our setting of continuous time random walk (see the next remark), they showed that a.s. for all ε > 0, there exists N 0 < ∞ a.s. such that for all N ≥ N 0 , Let us also mention the fact that Theorem 2 of [Rév04] states that for all ε > 0, almost surely we have sup x∈VN ℓ τN x − 2g log N ≥ (2(d − 4)/(d − 2) − ε) log log N for infinitely many N . This is not in contradiction with our Theorem 1.2.3 because we only give the typical behaviour (i.e. at a fixed time) of sup x∈VN ℓ τN x − 2g log N .
Remark 1.2.1. We have stated our results in the case of continuous time random walk but they hold as well for discrete time random walk. Unlike in the two-dimensional case, we have to do some modifications. The reason for this is because in dimension two we were essentially comparing exponential (continuous time) or geometrical (discrete time) variables with mean g log N to ag(log N ) 2 for some g > 0 and a ∈ (0, 1). In both cases, if we divide these variables by g log N then they converge to exponential variables with parameter 1. Thus there is no difference between the continuous time case and the discrete time one. In higher dimensions, we are comparing exponential or geometrical variables with mean g to ga log N and these two distributions have slightly different behaviour. More precisely, in the discrete time case we have to change the following points. In the definition of the measures µ a N the variables E x are now geometric variables with success probability 1/g which corresponds to the probability for the walk to never come back to its starting point. The description of the limit measures ν a in Theorem 1.2.1 is now different: the ℓ-component is a geometric distribution with the same success probability. Finally, we have to replace g by −1/ log(1 − 1/g) in Theorems 1.2.2 and 1.2.3.

Organisation of the paper and literature overview
Section 3 will be dedicated to the dimension two whereas Section 4 will deal with the dimensions greater or equal to three. Let us first describe the two dimensional case.
We first recall the definition of the GFF on the square lattice. With the notations of Theorem 1.1.2 in the square lattice case, the Gaussian free field is the centered Gaussian field φ N , indexed by the vertices in V N , whose covariances are given by the Green function: See [Ber16], [Zei12] for introductions to the GFF. Our argument will simply relate the thick points of the random walk to those of the GFF: see [Kah85], [HMP10] in the continuum and [BDG01], [Dav06] in the discrete case.
We now explain the interest of exploiting the connection to the GFF. As usual, the proofs of Theorems 1.1.1 and 1.1.2 rely on the method of (truncated) second moment. That is, a first moment estimate on |M N (a)| gives us the upper bound, while a matching upper bound on the second moment of |M N (a)| would supply the lower bound. Moreover, it is necessary to first consider a truncated version of |M N (a)|, where we consider points that are never too thick at all scales (this is similar to the idea in [Ber17]). Computing the corresponding correlations is not easy with the random walk, but is essentially straightforward with the GFF as this is basically part of the definition. As only an upper bound on the second moment is needed, comparisons to the GFF with Dynkin-type isomorphisms go in the right direction. We will see that the Eisenbaum's version will be the most convenient to work with.
We now state this isomorphism. Consider Γ = (V, E) a non-oriented connected infinite graph without loops, not necessary planar, and consider a walk Y on Γ. As in the isoradial case, we denote ℓ t x , x ∈ V, t ≥ 0, its local times, x 0 a starting point, V N the ball of radius N and center x 0 , τ N the first exit time of V N . We also denote by P x the law of Y starting from x and we assume that the following expression is symmetric in x, y: This allows us to define a centered Gaussian field φ N whose covariances are given by the previous expression. φ N is called Gaussian free field. We will denote P its law. The following theorem establishes a relation between the local times and the GFF (see lectures notes [Ros14] for a good overview of this topic) Theorem 2.0.1 (Eisenbaum's isomorphism). For all s > 0 and all measurable bounded function f : Remark 2.0.1. It would have been possible to use the generalized second Ray-Knight theorem (see [Ros14]). Compared to Theorem 2.0.1 above, this has the advantage that the laws of the GFFs on the left hand side and right hand side are the same. However this has an annoying drawback: indeed it is necessary to stop the walk where it starts, i.e. at x 0 . This isomorphism then leads to a GFF pinned at x 0 , i.e. is equal to 0 at x 0 and has free boundary condition. This is essentially equivalent to adding a global noise to a Dirichlet GFF of order √ log N which is sufficient to ruin second moment approach. This noise would have to be removed by hand in order to apply the method of second moment. This is possible but makes the proof substantially longer.
The Eisenbaum isomorphism immediately implies that √ ℓ τN x is stochastically dominated by |φ N (x) + s| / √ 2 with the right laws. The generalized second Ray-Knight theorem implies something similar but with differences as discussed above. One can actually show a stronger result and replace the absolute value on the right hand side by max(·, 0) (Theorem 3.1 of [Zha14]). Abe [Abe15] exploited this and used the symmetry of the GFF to make links between what was called thin points and thick points of the random walk on the 2-dimensional torus, up to a multiple of the cover time.
Organisation -planar case: The two-dimensional part of the paper will be organized as follows. In Section 3.1 we will present the general framework we deal with (Theorem 3.1.1). We will then show that Theorems 1.1.1 and 1.1.2 are simple corollaries. The upper bound, which is the easy part, will be briefly proved at the end of the same section. Section 3.2 is devoted to the lower bound. We first show that the probability to have a lot of thick points does not decay too quickly. This is the heart of our proof and makes use of the comparison to the GFF. We then bootstrap this argument to obtain the same statement with high probability, see Lemma 3.2.1 at the beginning of Section 3.2. This lemma is a key feature of our proof and allows us to use the comparison to the GFF. Indeed, since we do not require very precise estimates, we can deal with the change of measure coming from the isomorphism through very rough bounds, such as: |φ N (x 0 )| ≤ (log N ) 2 with high probability (see Lemma 3.2.2). This only introduces a poly-logarithmic multiplicative error in the estimate of the probabilities that two given points are thick, and so does not matter for the computation of the dimension of the number of thick points on a polynomial scale.
If we want more accurate estimates, more ideas are required. For instance, for the simple random walk on the square lattice, the comparison between the number of thick points for the random walk and for the GFF breaks down: the two following expectations converge as N goes to infinity: In the article [BL16] the thick points of the discrete GFF φ N were encoded in point measures of a similar form as the one we defined in (3). The authors showed the convergence of such measures. As a consequence, they went beyond the estimate (9) and showed that converges in law to a nondegenerate random variable. Question: In the case of simple random walk on the square lattice starting at the origin, converge to a nondegenerate random variable as N goes to infinity? Notice that the renormalisations are different in (10) and in (11). These differences suggest scraping the GFF approach if we want optimal estimates. This is what we will do in higher dimensions.
We have finished to discuss the two-dimensional case and we now describe the situation in higher dimensions. The article [DPRZ00] studied the thick points of occupation measure of Brownian motion in dimensions greater or equal to three. They obtained the leading order of the maximum and computed the Hausdorff dimension of the number of thick points. The article [CFR + 05b], as well as [CFR05a], [CFR06], [CFR07a], [CFR07c] (again, see [CFR07b] for a survey on this series of paper), studied the case of symmetric transient random walk on Z d with finite variance. One of their results computed the leading order of the maximum of the local times too. In both [DPRZ00] and [CFR + 05b], a key feature of the proofs is a localisation property (Lemma 3.1 of [DPRZ00] and Lemma 2.2 of [CFR + 05b]) which roughly states that a thick point accumulates most of its local time in a short interval of time. This property allows them to consider independent variables and makes the situation simpler compared to the two-dimensional case.
Let us also mention the paper [CCH15] which studied the scaling limit of the discrete GFF in dimension greater or equal to three. The authors obtained a result similar to Theorem 1.2.1. Namely, they showed that in the limit the field behaves as independent Gaussian variables. More precisely, they defined a point process analogue to ν 1 N (see (3)) which encodes the thick points of the GFF at criticality. They showed that this point process converges to a Poisson point process. Their situation is simpler because the intensity measure is governed by the Lebesgue measure rather than the occupation measure of Brownian motion. In particular, they could use the Stein-Chen method which allowed them to consider only the two first moments.

Organisation -higher dimensions:
Let us now present the main lines of our proofs and the organisation of the paper. In Section 4.1, Theorems 1.2.1, 1.2.2 and 1.2.3 will all be obtained from the joint convergence of the sequences of real-valued random variables ν a N ( We will obtain this fact by computing explicitly all the moments of these variables (Proposition 4.1.1). This is actually the heart of our proofs and Section 4.2 will be entirely dedicated to it. To compute the k-th moment of ν a N (A × T ), we will estimate the probability that the local times in k different points, say x 1 , . . . , x k , belong to 2ga log N + T . In the subcritical regime (a < 1), we will be able to assume that these points are far away from each other. In that case, Lemma 4.2.2 will show that we can restrict ourselves to the event that there exists a permutation σ of the set of indices {1, . . . , k} which orders the vertices so that we have the following: the walk first hits x σ(1) , accumulates a big local time in x σ(1) , then hits x σ(2) accumulates a big local time in x σ(2) , etc. When the walk has visited x σ(i) it does not come back to the vertices x σ(1) , . . . , x σ(i−1) . The local times can thus be treated as if they were independent.
At criticality (a = 1), we do not renormalise the number of thick points and we will a priori have to take into account points which are close to each other. Here, the key observationcontained in Lemma 4.2.3 and already present in Corollary 1.3 of [CFR + 05b] -is that if two distinct points are close to each other, then the probability that they are both thick is much smaller than the probability that one of them is thick, even if they are neighbours! This is specific to the dimension greater or equal to 3 and tells that the thick points do not cluster. Thus, only the points which are either equal or far away from each other will contribute to the k-th moment. Section 4.3 will contain the proofs of four intermediate lemmas that are needed to prove Proposition 4.1.1 on the convergence of the moments of ν a

General framework and upper bound
We now describe the general setup for the theorem. Consider Γ = (V, E) a non-oriented connected infinite graph without loops, not necessary planar, and (Y t ) t≥0 a continuous time random walk on Γ, not necessary a nearest neighbour walk. As before, we take x 0 ∈ V a starting point and write d Γ for the graph distance. We will also write starting from x ∈ V and E x the associated expectation. We introduce the first exit time of V N and the local times: Finally we will denote G N the Green function, i.e.: Notation: For two real-valued sequences (u N ) N ≥1 and (v N ) N ≥1 and for some parameter α, We now do the following assumptions on the graph Γ and on the walk Y :

Assumptions
To ensure the existence of the GFF, we will need to assume: We now make two types of assumptions: the first one concerns the Green function and the second one is about the density of the graph Γ. We assume that #V N (x 0 ) = N 2+o(1) and that for all and where we control the Green function as follows: Assumption 2. There exists g > 0 such that: which can be thought of as a circle of radius R centered at x: Finally, we assume that the jumps are not unreasonable: . We now briefly discuss the above assumptions. Note that we have assumed that all the bounds do not depend on the starting point x ′ 0 ∈ V N (x 0 ). This will be important for our Lemma 3.2.1. Assumptions 1 and 2 may first require to change the holding times of the walk. Assumption 3 is needed to go beyond the L 2 phase whereas Assumption 4 is needed to bootstrap the probability to have a lot of thick points (Lemma 3.2.1). This latter assumption can be weakened. We could replace which goes to zero quickly enough as t goes to zero. For instance, any positive power of t would do.
As confirmed by the theorem below, a sensible definition of a-thick points is given by Theorem 3.1.1. Assuming the above assumptions we have the following two P x0 -a.s. convergences: We now check that Theorems 1.1.1 and 1.1.2 are consequences of this last theorem, i.e. we check that these two setups satisfy Assumptions 1 -4 above. In the setting of Theorem 1.1.1, the reversibility of the chain is ensured by the symmetry of the increments, while it is automatic in the setting of Theorem 1.1.2. Also, in the latter setting, the walk is a nearest-neighbour random walk so Assumption 4 is clear. The following lemma finishes to prove that all the assumptions are fulfilled if in both cases we take Moreover for all η ∈ (0, 1), 2. Isoradial Graphs. Consider a walk Y as in Theorem 1.1.2. Then for all η ∈ (0, 1), Proof. Square lattice. We first start to prove (17). By translation invariance, we can assume x ′ 0 = 0. We consider the discrete time random (S i ) i≥0 associated and we are going to abusively write τ N to denote the first time the discrete time walk exits V N . Take λ > 0 to be chosen later on. The probability we are interested in is not larger than As the increments have a finite variance, the first term on the right hand side is not larger than CλN 2 /M 2 for some C > 0 by an union bound. Secondly, Theorem 2.3.9 of [LL10] gives estimates on the heat kernel and in particular implies that there We obtain (17) which is linked to the Green function by: where the lower bound (resp. upper bound) is satisfied by all (1), we are thus left to show that the elements z such that d Γ (x 0 , z) > N (log N ) 2 do not contribute to the sum in the equation (22). Thanks to (17), we have which goes to zero as N goes to infinity. It concludes the square lattice part of the lemma.
Isoradial graphs. (20) and (21) are a direct consequences of Theorem 1.6.2 and Proposition 1.6.3 of [Law96] in the case of simple random walk on the square lattice. Kenyon extended this result to general isoradial graphs (see [Ken02] or Theorem 2.5 and Definition 2.6 of [CS11]).
From now on, we will work with a graph Γ and a walk Y which satisfy assumptions 1 -4. An upper bound on the Green function G N is already enough to prove the upper bound of Theorem 3.1.1: Proof of the upper bound of Theorem 3.1.1. Let a ≥ 0 and N ≥ 1. For every ε > 0 we obtain by Markov inequality: But for every x ∈ V N , under P x , ℓ τN x is an exponential variable with mean G N (x, x). Hence by (14a), The upper bound for the convergence in probability follows. To show that we observe that, taking N = 2 n in (23), decays exponentially and so is summable. Moreover, if 2 n ≤ N < 2 n+1 , Hence the Borel-Cantelli lemma implies that

Lower bound
We first start this section by establishing a lemma which simplifies a bit the problem: we only need to show that the probability to have a lot of thick points decays sub-polynomially. For all starting point with p N = p N (a) > 0 decaying slower than any polynomial, i.e. log p N = o a,ε (log N ). Then for all a ∈ (0, 1),

Proof.
A similar but weaker statement appears in [DPRZ01] and [Ros05] where they assumed that p N was bounded away from 0. The idea is to decompose the walk on the ball V N (x 0 ) into several walks on smaller balls to bootstrap the probability we are interested in. First of all, let us remark that if p N ∈ (0, 1) decays slower than any polynomial, then so does (inf n≤N p n ) N ≥1 . Consequently, we can assume without loss of generality that the sequences p N in the statement of the lemma are non increasing.
Fix ε > 0 and take N large and K N ∈ N much smaller than N such that K N = N 1−o(1) . Let us introduce the stopping times . So by a repeated application of Markov property, we see that for all δ > 0, if N is large enough so that a(log N ) 2 ≤ (a + δ)(log K N ) 2 (which is possible by assumption on K N ), we have: (24) To conclude, we have to choose K N small enough to ensure that i max is large with high probability. If the walk was a nearest neighbour random walk, we could say that i max + 1 ≥ ⌊N/K N ⌋ P x0 -a.s. Here, the jumps may be unbounded but large jumps are costly (assumption (16)) so we will be able to recover a lower bound fairly similar on i max . By the triangle inequality, we have for all k ≥ 1 Assumption (16) allows us to bound this last probability: there exists (ε N ) N ≥1 ⊂ (0, ∞) which converges to zero such that if M > 0, Coming back to the estimate (24) and taking k = (log N )/p N , we have obtained We can choose so that the previous estimates gives We now conclude as in the proof of the upper bound of Theorem 3.1.1. We apply the Borel-Cantelli lemma along the sequence (2 p ) p∈N which yields This finishes the proof of the lemma because log 2 p+1 / log (2 p ) → 1 as p → ∞.
As mentioned at the end of Section 2, when we will use Eisenbaum's isomorphism, we will have to bound from above expectations of the form: for some given event A. We will use the following elementary lemma which we state here only for convenience:

Lemma 3.2.2. For all events A, for all N large enough
Proof. Using (14a), we have: which concludes the lemma.
We now provide our proof of the lower bound of Theorem 3.1.1. In the following, we write our arguments with the starting point x 0 but note that the same also works for all starting points , what is required to apply Lemma 3.2.1. Proof of the lower bound of Theorem 3.1.1. During the entire proof we will fix some small η > 0. To ease notations, we will denote Q N := Q N (x 0 ). Recall that if x ∈ Q N and 1 ≤ R ≤ N 1−η , assumptions (15) give the existence of a subset C R (x) ⊂ Q N which can be thought of as a circle of radius R around x. We will denote M x R the operator corresponding to taking the mean value of a function on this circle: if f is a function defined on Q N , then We use Eisenbaum's isomorphism with some s > 0 (s = 1 will do). Let ε N = 1/ √ log N and for some b > a (to be chosen later on, close to a) and φ N a GFF independent of the walk, we define the good events at x: We require the points to be never to thick at any scales (similar to [Ber17]). We restrict ourselves to Q N (the subset of V N where we control G N ) by considering: and it remains to estimate the first and second moments on the right hand side.

First Moment Estimate
Firstly, we estimate the first moment without restricting to any event. Thanks to assumptions (14b) and (14c) and because, starting from x, the law of ℓ τN x is exponential, we have: To estimate the probability P (G η N (x, φ N )) we will first derive a large deviation estimate for M x R (φ N + s) 2 . The estimate we obtain is rough and does not take into account the fact that if R is large we should expect M x R (φ N + s) 2 to be close to its mean. Writing N (µ, σ 2 ) a Gaussian variable with mean µ and variance σ 2 , by Jensen's inequality we have ∀λ > 0 and ∀t ∈ (0, 1/(2g)) where 0 < C(t) < ∞ because tg is smaller than 1/2. Hence, we have obtained: for all t ∈ (0, 1/(2g)), there exists C(t) ∈ (0, ∞) such that Hence, using the above estimate with t = 1/(4g) for instance, if x ∈ Q N , the probability that the good event at x linked to φ N does not hold is: for some C(η) > 0. By independence of φ N and the local times of the random walk, we thus have Now, using the Eisenbaum's isomorphism and Lemma 3.2.2, we can bound from above the probability P x0 ℓ τN By taking δ = 2 a/g, we can bound from above the probability appearing in the last sum by: where P is the shifted probability: By Cameron-Martin theorem, under this new probability, φ N has the same covariance structure but the mean of φ N (y) is now given by: As we have taken b > a, we can apply our tail estimate (26) to show that, for some small t > 0 which may depend on η, a and b. With the estimate on the first moment without the event G b,η N , this shows that:

Second Moment Estimate
To control the second moment, we adapt the ideas of [Ber17] to our framework: let x, y ∈ Q N such that d Γ (x, y) ≤ N 1−η . We can find some R ∈ (2 p ) p∈N , R ≤ N 1−η such that As before, we apply the Eisenbaum isomorphism, Lemma 3.2.2, an exponential Markov inequality, and using the fact that by Cauchy- where P denotes the shifted probability defined by By Cameron-Martin theorem, under the probability P, φ N has the same covariance structure but the mean of φ N (z) is now given by: by our particular choice of R. Thanks to assumptions (14b) and (15b), one can check that the .
Again thanks to our particular choice of R, we have obtained: As a < 1, we can choose b > a close enough to a to ensure that the exponent 4a − 2(2 √ a − √ b) 2 is less than 2. We can then sum over all x, y ∈ Q N such that |x − y| ≤ N 1−η and use assumption (13) to find that: We eventually treat our last sum noticing that the probability in this sum is not larger than (using (27) without the termP(· · · )): This shows that the second moment is not larger than N 4(1−a+aη)+oη(1) . To come back to the probability we wanted to bound from below, this implies: As this is true for all η > 0, it means that the probability is not less than (1/N ) o(1) . We can then use Lemma 3.2.1 to conclude the proof of Theorem 3.1.1. 1.2.1, 1.2.2 and 1.2

.3
This section is devoted to the proofs of Theorems 1.2.1, 1.2.2 and 1.2.3. Let us first recall the setting and introduce some new notations. Consider a continuous time random walk (Y t ) t≥0 on Z d for d ≥ 3 and denote P x and E x its law and expectation starting from x. Writing V N = {−N, . . . , N } d , we consider the first exit time of V N and the first hitting time of x: We will denote G and G N the Green function on Z d and on V N respectively: for all x, y ∈ Z d , Finally, we denote g : = G(0, 0) and for allx,ỹ ∈ (−1, 1) d , we have the following pointwise estimate: The proof of this lemma will be given in Section 4.3. As mentioned in Section 2, a key point is to show that all the moments of the number of thick points converge which is the purpose of the next proposition. Before stating it, let us introduce some notations.
Notations: If k ≥ 1 and q ≥ 1, we denote by f (k → q) the number of ways to partition a set with k elements into q non empty sets. As this is equal to the number of surjective functions from {1 . . . k} to {1 . . . q} divided by q!, we have If X is a topological space we will denote by B(X) the class of Borel sets of X.
Moreover, we assume that the A i × T i 's are pairwise disjoint. By denoting k = k 1 + · · · + k r we define Ti with the convention y σ(0) = 0. 1. Subcritical regime: let a ∈ [0, 1) and if a = 0 assume furthermore that T i ⊂ (0, ∞) for all i. Then

At criticality,
The previous results also hold if we replace ν a N by µ a N .
We postpone the proof of this proposition to the next section and we now explain how we can deduce Theorems 1.2.1, 1.2.2 and 1.2.3 from it. We start with Theorem 1.2.1.
Proof of Theorem 1.2.1. This proof will be decomposed in three small parts. First, we will show that the previous proposition implies the joint convergence of (ν a N (A 1 × T 1 ), . . . , ν a N (A r × T r )) with suitable A i 's and T i 's. The second part is relatively standard and shows that it then implies the convergence in law of the sequence of random measures {ν a N , N ≥ 1}. The third part is dedicated to the identification of the limiting measures.
Step 1. Take a ∈ [0, 1]. Let us first show that the previous proposition implies the convergence of the joint distribution (ν a N (A 1 × T 1 ), . . . , ν a N (A r × T r )) where the A i 's and T i 's are as in the statement of the proposition. As all their moments converge, we just need to check that the limiting moments do not grow too rapidly. Take k 1 . . . k r ≥ 1. We notice that for all for some universal constant C depending only on the dimension d. Hence there exists C ′ depending on d and on the T i 's such that with k = k 1 + · · · + k r . In particular, it implies that the moment generating function associated to those moments has a positive radius of convergence and they determine a unique law. It thus proves the claimed convergence in the subcritical regime. At criticality, we notice that for all is not larger than the number of ways to partition a set of k elements into no more than q parts which is equal to q k /(q!). Using (35), it implies that Again the radius of convergence of the associated moment generating function is positive and it gives the required convergence in the critical case as well. We will denote ν a (A 1 ×T 1 ), . . . , ν a (A r × T r ) random variables which have the limiting distribution of (ν a N (A 1 × T 1 ), . . . , ν a (A r × T r )).
Step 2. We now show the convergence of the sequence of random measures {ν a N , N ≥ 1}. Recalling that the underlying topology is the topology of vague convergence, it is enough to show that for all function φ : converges in distribution. It is enough to check that for all L-Lipschitz function h : R → R, E 0 [h( ν a N , φ )] converges. By Lemma 4.3.2, we can uniformly approximate φ by a sequence of functions (φ p ) p≥1 taking the following form: By the joint convergence proven in Step 1, for all p ≥ 1, = ν a , φ p and we can define the law (by dominated convergence theorem for instance) We are going to show that we can exchange the two limits, i.e. that ν a N , φ converges in law to By the first part of the proof, the first term goes to zero as N goes to infinity. If t 0 ∈ R is such that the support of φ is included in [−1, 1] d × (t 0 , ∞), then the second term is not larger than Thus the limit of the second term goes to zero when p → ∞. The third term goes to zero by definition and we have proved Step 3. The convergence of the sequence of random measures {ν a N , N ≥ 1} has thus been proved. We are now going to identify the limit. What we did in Step 1 and Step 2 shows that the limiting distribution is entirely determined by the limiting moments from Proposition 4.1.1. In particular, the same conclusion holds for both {ν a N , N ≥ 1} and {µ a N , N ≥ 1} and this shows that these two sequences converge and have the same limiting distribution. We are now going to show that the limiting measures can be expressed in terms of the occupation measure µ occ and a Poisson point process as explained in Theorem 1.2.1. We start by the subcritical regime (a < 1). Take A i × T i , i = 1 . . . r, as in Proposition 4. 1.1, k 1 , . . . , k r ≥ 1 and denote k = k 1 + · · · + k r . As is the Green function associated to Brownian motion killed at the first exit time τ of [−1, 1] d (see equation (3.15) of [Bas95] for instance), it is not hard to see that with the convention y σ(0) = 0. Thus This proves the identification (4) of the limiting measure in the subcritical regime. Let us now consider the critical case a = 1. Recalling the definition of f in (31) we see that the equation (56) of Lemma 4.3.1 implies that if P 1 (λ 1 ), . . . , P r (λ r ) are independent Poisson random variables with parameters λ 1 , . . . , λ r , Using (36), this now shows (5) and it concludes the proof.

Proof of Proposition 4.1.1
In this section, we will prove Proposition 4.1.1 stated in the previous section. We are first going to lay the groundwork by stating some technical lemmas which will be used in the proof of Proposition 4.1.1. These lemmas, except the next one, will be proven in Section 4.3. We start with a well-known and easy lemma that we state for convenience. This lemma is valid for more general Markov chains.

Lemma 4.2.1. For all subset
x and Y τA 1 {τA<∞} are independent. Proof. Consider a trajectory of the random walk Y starting at x and killed at τ A . We can decompose it according to the excursions away from x. There is a geometric number of independent excursions. The last one is conditioned to not come back to x whereas the previous ones are i.i.d. excursions conditioned to come back to x. To conclude the proof, we notice that Y τA 1 {τA<∞} depends on the last excursion whereas ℓ τA x depends on the previous ones.
Remark 4.2.1. This lemma implies in particular that conditioned on Y τA 1 {τA<∞} and starting from x, ℓ τA x is still an exponential variable with mean E x [ℓ τA x ]. We also want to emphasize that this lemma is no longer true if the walk does not start at x. Now, consider the k-th moment of ν a N (A × T ). To compute it, we will have to estimate the probability that in k different points, say x 1 , . . . , x k , the local times belong to 2ga log N + T . To capture the correlations of those local times, we will denote E (to ease notation, we omit the dependence in N and x 1 , . . . , x k ) the number of excursions between the x i 's before the time τ N . More precisely, if we define such that the following is true. For all (y 1 , . . . , y k ) and (y ′ with the convention y 0 = y ′ 0 = 0. For all p ≥ k − 1 and all x 1 , . . . , x k non zero and pairwise distinct elements of V N , (40) Moreover, if x 1 = ⌊N y 1 ⌋ , . . . , x k = ⌊N y k ⌋, for y 1 , . . . , y k non zero and pairwise distinct elements of (−1, 1) d , we have the following pointwise estimate: with the convention y σ(0) = 0.
Remark 4.2.2. It is important for us to give a better estimate than As mentioned in Section 2, in the subcritical regime we will be able to restrict ourselves to points x 1 , . . . , x k which are far away from each other. At criticality we will have to deal with points which are close to each other. The following lemma shows that two distinct close points are not thick at the same time with high probability: under P x . If x = y, then for all p ≥ 1, there exists ε p > 0 independent of x and y such that for all t ∈ R, We have now all the ingredients we need to start the proof of Proposition 4.1.1.
Proof of Proposition 4.1.1. To ease notations, we will restrict ourselves to the case of the k-th moment of ν a N (A × T ) for some suitable A ⊂ [−1, 1] d and T ⊂ R. Indeed, the proof of the general case follows almost entirely along the same lines and throughout the proof we will explain which arguments need to be changed to treat the case of mixed moments When we will refer to the general case, k will denote k 1 +· · ·+k r . The proof will be in three parts. The two first ones will deal with the estimates on the moments of ν a N (A × T ) in the subcritical regime and at criticality respectively, whereas the third part will briefly show the results on µ a N .

Subcritical regime, ν a N .
We first start with the subcritical regime case by considering (a, T ) ∈ ((0, 1) × B(R)) ∪ ({0} × B((0, ∞))) with inf T > −∞ and A ∈ B([−1, 1] d ) such that the Lebesgue measure ofĀ\A • vanishes. In the following, we will take N large enough so that 2ga log N + T ⊂ (0, ∞). To ease notations, we will denote The k-th moment of M N can be written as In the general case, the sum is over (x 1 , . . . , x k ) ∈ (A 1N ) k1 × · · · × (A rN ) kr and for each group i = 1 . . . r, the local times are required to belong to 2ga log N + T i . We will prove that this moment is bounded by induction on k: given that the moment of order k − 1 is bounded, we deduce that the contribution of points x 1 , . . . , x k which are too close to each other or too close to 0 is negligible (this is the only argument which crucially uses the fact that a < 1). More precisely, for some r N = N o(1) (to be chosen later on), we introduce and we claim that the contribution of the points (x 1 , . . . , x k ) ∈ (A N ) k \A N,k converges to zero when N goes to infinity. Indeed, this contribution is at most which goes to zero: this is clear for k = 1 and comes from the induction hypothesis for k ≥ 2. We have proved that: For a given x ∈ V N \∂V N , the Lebesgue measure of the set {y ∈ (−1, 1) d : ⌊N y⌋ = x} is (1/N ) d .
Hence we can write We will first bound from above the integrand. This will provide us the domination we need in order to apply the dominated convergence theorem and we will be left to show the pointwise limit.
Let (x 1 , . . . , x k ) ∈ A N,k . By definition of E (equation (37)), if the walk visits all the x i 's before τ N , then E ≥ k − 1. Thus In this paragraph, we will use Lemma 4.2.2 to show that the probability P 0 ℓ τN x1 , . . . , ℓ τN x k ∈ 2ga log N + T, E ≥ k is very small. First, by denoting t := inf T /g, we can bound Starting from x 1 , the law of the time spent in x 1 before hitting ∂V N ∪ {x 2 , . . . , x k } is an exponential law with mean at most g. Also, if E = p, the number of excursions from x 1 to {x 2 , . . . , x k } before τ N is not larger than p. Hence, by Lemma 4.2.1 conditioned on the event {E = p, τ xi < τ N ∀i ≤ k}, the joint law (ℓ τN x1 , . . . , ℓ τN x k ) is stochastically dominated by the law of k independent Gamma random variables with shape parameter p + 1 and scale parameter g. Using the claim (57) of Lemma 4.3.1 about the Gamma distribution, it implies that Let U (x 1 , . . . , x k ) be as in Lemma 4.2.2. Then goes to zero and we have obtained: According to Lemma 4.2.2, the function (y 1 , . . . , y k ) ∈ (−1, 1) k → U (y 1 , . . . , y k ) ∈ (0, ∞) is integrable. Moreover, the equation (39) of Lemma 4.2.2 implies that if y 1 , . . . , y k ∈ (−1, 1) d are such that (⌊N y 1 ⌋ , . . . , ⌊N y k ⌋) ∈ A N,k , then Our last task consists in controlling the probability appearing in the equation (46). By Lemma 4.2.1, conditioning on the event {E = k − 1, τ xi < τ N ∀i = 1 . . . k}, the local times ℓ τN xi , i = 1 . . . k, are independent exponential variables with mean E xi ℓ τN ∧min j =i τx j xi ≤ g. Consequently, Using the first estimate of Lemma 4.2.2, it implies that E 0 (M N ) k is bounded and it also provides us the domination we need to use the dominated convergence theorem. We have already done everything we need for the pointwise convergence. Indeed, if x 1 = ⌊N y 1 ⌋ , . . . , x k = ⌊N y k ⌋, for y 1 , . . . , y k non zero and pairwise distinct elements of (−1, 1) d , Lemma 4.2.2 provides an explicit expression for the pointwise limit and a small modification of the arguments in the proof of Lemma 4.2.2 shows that Moreover, Notice the interior A • and the closureĀ in the previous inequalities. As we have supposed that the Lebesgue measure ofĀ\A • vanishes, putting things together leads to the convergence of with the convention y σ(0) = 0. This is exactly (33) in the case r = 1. In the general case of a mixed moment, we recover the result by the exact same method.

At criticality, ν a N .
Let us now consider the critical case a = 1. Again take T ∈ B(R) with inf T > −∞ and A ⊂ [−1, 1] d such that the Lebesgue measure ofĀ\A • vanishes. As mentioned before, the only argument which cannot be reproduced from the subcritical case is the very first one because the contribution of the points (x 1 , . . . , x k ) ∈ (A N ) k \A N,k is not negligible. In particular, (x1,...,x k )∈A N,k P 0 ℓ τN x1 , . . . , ℓ τN x k ∈ 2g log N + T converges again to m(A × T, k) (defined in (32)). Let us first notice that the problems come from the points which are close to each other and do not come from points at the origin. In other words, the points (x 1 , . . . , x k ) ∈ (A N ) k with one of the x i 's equal to zero do not contribute. Indeed, by ignoring the points which are within a distance 2r N to each other or to zero, which contributes at most Cr d N for every such point, we have: The last sum is over l different points and we require the local times to be large in l + 1 different points. We can then use the same arguments as in the subcritical regime (all the points are far away from each other) to show that this last sum is at most CN −2 . As r N = N o(1) it genuinely shows that this contribution vanishes.
We are going to estimate If (x 1 , . . . , x k ) ∈ (A N \{0}) k \A N,k , by definition of A N,k , it means that there are at least two balls B(x i , r N ) which overlap. In the following, we will partition the set (A N \{0}) k \A N,k according to the maximum number r (r ≤ k − 1) of balls which do not overlap. We will denote by x ip , p = 1 . . . r, the centers of such balls and we will partition the set of indices ⊔ r p=1 I p = {1, . . . , k} such that for all p = 1 . . . r, i ∈ I p , x i − x ip ≤ 2r N . See Figure 2. The reader should think of the balls as small balls which are far away from each other. The choice of the partition (I p ) may be not unique. In this case, we make an arbitrary choice.
Our decomposition is thus: By denoting t := inf T /g, we can first bound: If (x 1 , . . . , x k ) ∈ W = N,k,r,(Ip) , then there exists p 0 ∈ {1, . . . , r} and j p0 ∈ I p0 such that x ip 0 = x jp 0 . To bound from above this last sum, for each p = p 0 we keep track of only one x k , k ∈ I p , by considering x ip . As for all k ∈ I p , x k − x ip ≤ 2r N , our estimate is increased by a multiplicative factor of order r d N for each point that we forget. For p = p 0 , we keep track of both x ip 0 and x jp 0 . Furthermore, x jp 0 will absorb all the x ip , p = p 0 which are within a distance 2r N of x jp 0 . This procedure implies that: where C > 0 may depend on d, k, r. We will conclude by showing that this last sum is not larger than N −ε for some ε > 0. Take Hence if p max is large enough, the negative power (p max − s)(2 − d) + d of r N will kill the positive power (k − s − 1)d of r N in the equation (49) We have obtained the existence of ε > 0 such that where we justify as before the last inequality thanks to the integrability of U and by (39). This concludes the proof of the estimates on E 0 {ν a N (A × T )} k at criticality (equation (34) with r = 1).
In the general case of a mixed moment, we have to deal with points As before, we decompose this set according to blocks of points which are close to each other. Again, only points which are equal inside a same block will contribute. As we have assumed that the A i × T i 's are pairwise disjoint, they will not interact between each other meaning that We notice that the number of ways to partition the sets {1, . . . , k i } into r i non empty sets, for i = 1 . . . r, is equal to Thus, the contribution of points (x 1 , . . . , This shows (34) in the general case r ≥ 1.

Estimates on µ a N .
We now briefly end the proof of Proposition 4.1.1 by explaining how the results for µ a N are obtained. Take a ∈ [0, 1], T ∈ B(R) and A ⊂ [−1, 1] d such that the Lebesgue measure ofĀ\A • vanishes. By definition of f (k → r) and since (E x ) x∈VN are i.i.d. exponential variables with mean g independent of M N (0), the normalised k-th moment E 0 (µ a N (A × T )) k is equal to We have already shown that

Proof of technical lemmas
We start this section by proving Lemma 4.1.1 which gives estimates on the Green function G N (defined in (29) as well as the Green function G on Z d ) in dimension greater of equal to 3.
Proof of Lemma 4.1.1. As in dimension 2, these estimates follow from [Law96] and [LL10]: Proposition 1.5.8 in [Law96] gives and Theorem 4.3.1 in [LL10] (or Theorem 1.5.4 in [Law96] for a slightly worse estimate) gives Our two first estimates on the Green function on the diagonal follow since if y ∈ V (1−η)N for some η > 0, then for all z ∈ ∂V N , |z − y| ≥ ηN . The lower bound on q N (x, y) follows as well.
We now move on to the proof of Lemma 4.2.2. We consider k non zero and pairwise distinct points x 1 , . . . , x k ∈ V N and we recall the definitions of E and of the stopping times ς p in (37).
Proof of Lemma 4.2.2. As mentioned just before Lemma 4.2.2, if E = k − 1 and τ xi < τ N ∀i = 1 . . . k then the stopping times ς p , p = 0 . . . k − 1, define a permutation σ of the set of indices {1, . . . , k} which keeps track of the order of visits of the set {x 1 , . . . , x k }. By a repeated application of Markov property, we thus have: But for all σ ∈ S k and i = 1 . . . k − 1, We bound from below the denominator G N (x σ(i+1) , x σ(i+1) ) by 1 and from above the numerator with the convention x σ(0) = 0. The general case p ≥ k − 1 follows from the same lines but now the order of visits of the set {x 1 , . . . , x k } is not as simple as before. In the following, σ ∈ S k will keep track of the order of new visits of the vertices x 1 , . . . , x k : x σ(1) is the first vertex visited among the x i 's, x σ(2) the second one... We will focus on the transitions which explore new vertices, so we introduce the notion: (σ, f ) ∈ S k × {1, . . . , k} {2,...,k} is said to be admissible if ∀i = 2 . . . k, f (i) ∈ {σ(1), . . . , σ(i − 1)}.
x f (i) will denote the vertex visited just before visiting the vertex x σ(i) . Now we define U (x 1 , . . . , x k ) := (σ,f ) admissible By keeping track of the transitions where new vertices are discovered (in a chronological sense) and by noticing that all the others occur with a probability which is not larger than C k max i =j |x i − x j | 2−d , we have This proves (40). We notice that (39) is immediate from the definition of (y 1 , . . . , y k ) ∈ (−1, 1) k → U (y 1 , . . . , y k ) and we now check that it is integrable. Take (σ, f ) admissible. There is only one occurrence of y σ(k) in the product, so we can first integrate: We then proceed inductively by integrating next with respect to y σ(k−1) , and so on. This proves that U is integrable. We now turn to (41). If x 1 = ⌊N y 1 ⌋ , . . . , x k = ⌊N y k ⌋, for y 1 , . . . , y k non zero and pairwise distinct elements of (−1, 1) d , then there exists η ∈ (0, 1) such that for all N large enough, x i ∈ V (1−η)N , |x i | ≥ ηN and for all i = j, |x i − x j | ≥ ηN . Hence Lemma 4.1.1 implies which leads to: a d g |y 1 − y 2 | 2−d − q(y 1 , y 2 ) .
We now prove Lemma 4.2.3.
By decomposing the walk along the different excursions between x and y, by Lemma 4.2.1 we see that starting from x the joint law of ℓ ∞ x , ℓ ∞ y can be stochastically dominated by: where A is a geometric random variable with failure probability (p xy ) 2 = P x (∃0 < s < t, Y s = y, Y t = x) and ℓ x,j , ℓ y,j , j ≥ 1, are i.i.d. exponential variables with mean θ xy independent from A. A is the number of round trips between x and y and ℓ x,j is the time spent in x during the j-th round trip. Let us mention that it is not an exact equality in distribution but only a stochastic domination. Indeed, we exactly have: starting from x, but the number of ℓ y,j 's we have to sum up is A (resp. A − 1) if the last visited vertex is y (resp. x). However this stochastic domination is sufficient for our purposes. Let p ≥ 0. For all i = 1 . . . p + 1 we stochastically dominate as above ℓ ∞,i x , ℓ ∞,i y by variables with a superscript i and we have Conditioned on the value of A i j=1 ℓ i y,j are two independent Gamma variables. We can thus use the claim (57) of Lemma 4.3.1 and We are going to bound from above the last sum indexed by n. Let us first notice that p xy and θ xy are linked by a simple formula. Indeed, (54) implies that E x [ℓ ∞ x ] = E [A] E [ℓ x,1 ], meaning that g = θ xy / 1 − p 2 xy . Then inf x =y g(1 − p xy )/θ xy = inf x =y 1/(1 + p xy ) > 1/2 so we can find λ > 1 such that inf x =y g(1 − λp xy )/θ xy > 1/2. If the index q in the equation (55) is large enough, say q ≥ q 0 (p), then for all n ≥ ⌈q/2⌉ − p we have 2 log(λ)n ≥ p log(n + p) and we can bound If q < q 0 (p), we bound the sum indexed by n by some constant depending on p. Overall, coming back to the equation (55), we can further bound from above the probability we are interested in by: We have chosen λ to make sure that the previous exponent is smaller than −2 which is exactly what was required.
We now state and prove elementary Lemma 4.3.1 (recall the definition of f (k → q) in (31)).