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.
For , consider a continuous time simple random walk on with rate 1. Let us denote the law of starting from and the associated expectation. Defining , we denote the first exit time of and 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 . Namely, we show that the field behaves like the field composed of i.i.d. exponential variables with mean 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 .
1.1 Dimension two
Consider a continuous time random walk on starting from the origin where is the jump process with i.i.d. increments and is an independent Poisson process of parameter . As before, we consider the square of side length , the first exit time of and the local times defined as in (1). The theorem below shows that corresponds to the maximum, and for , we call the set of -thick points
Then we have the following:
Assume that the law of the increments is symmetric (i.e. ), centered, with a finite variance and denote the covariance matrix of the increments. Then we have the following two a.s. limits:
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 be any connected infinite isoradial graph, with common radius 1, i.e. is embedded in and each face is inscribed into a circle of radius 1. Note that if are adjacent then and , together with the centers of the two faces adjacent to the edge , form a rhombus. We denote by the angle at (or at ). See the figure 1 for an example. For instance, the square (resp. triangular, hexagonal, etc) lattice is an isoradial graph with (resp. , etc) for all . We assume the following elliptic condition:
Define the conductance and let be a Markov jump process with conductances . is a continuous time walk which waits an exponential with mean time in each vertex and then jumps from to with probability . Take a starting point and denoting the graph distance we define for all
and as before (equation (1)), we consider the first exit time of and the local times. We will denote the law of the walk starting from and the associated expectation.
As confirmed by the theorem below, a sensible definition of -thick points is given by
We have the following two -a.s. limits:
Theorems 1.1.1 and 1.1.2 also hold when we consider the walk up until a deterministic time, say, rather than the first exit time of , since 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.
1.2 Higher dimensions
We now come back to the setting of the beginning of Section 1 for and we denote . In this section, the walk starts at the origin of .
We describe thick points through a more precise encoding by considering for the point measure:
Let us emphasize that the normalisation factor is equal to 1 when . We view as a random measure on . We compare the thick points of random walk with the thick points of i.i.d. exponential random variables with mean located at each visited site by the walk. More precisely, we denote and taking i.i.d. exponential variables with mean independent of , we define
We finally denote by the first exit time of of Brownian motion starting at the origin and by the occupation measure of Brownian motion starting at the origin and killed at . Then we have:
For all there exists a random Borel measure on such that, relatively to the topology of vague convergence of measures on (on if ), we have:
Moreover, for all the distribution of does not depend on and
At criticality, is a Poisson point process:
We will see that this statement will imply the following two theorems:
If we define for every the set of -thick points:
then there exist random variables such that for all
Moreover, for all the distribution of does not depend on and
is a Poisson variable with parameter : for all
There exists an almost surely finite random variable such that
Moreover, is a Gumbel variable with mode (location of the maximum) and scale parameter , i.e. for all
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 , there exists a.s. such that for all ,
Let us also mention the fact that Theorem 2 of [Rév04] states that for all , almost surely we have for infinitely many . 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 .
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 to for some and . In both cases, if we divide these variables by 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 to 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 the variables are now geometric variables with success probability which corresponds to the probability for the walk to never come back to its starting point. The description of the limit measures in Theorem 1.2.1 is now different: the -component is a geometric distribution with the same success probability. Finally, we have to replace by in Theorems 1.2.2 and 1.2.3.
2 Organisation of the paper and literature overview
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 , indexed by the vertices in , 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 gives us the upper bound, while a matching upper bound on the second moment of would supply the lower bound. Moreover, it is necessary to first consider a truncated version of , 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 a non-oriented connected infinite graph without loops, not necessary planar, and consider a walk on . As in the isoradial case, we denote , its local times, a starting point, the ball of radius and center , the first exit time of . We also denote by the law of starting from and we assume that the following expression is symmetric in :
This allows us to define a centered Gaussian field whose covariances are given by the previous expression. is called Gaussian free field. We will denote 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 and all measurable bounded function ,
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 . This isomorphism then leads to a GFF pinned at , i.e. is equal to at and has free boundary condition. This is essentially equivalent to adding a global noise to a Dirichlet GFF of order 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 is stochastically dominated by 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 (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: 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 goes to infinity:
In the article [BL16] the thick points of the discrete GFF 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, does
converge to a nondegenerate random variable as goes to infinity?
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 [CFR05b], 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 with finite variance. One of their results computed the leading order of the maximum of the local times too. In both [DPRZ00] and [CFR05b], a key feature of the proofs is a localisation property (Lemma 3.1 of [DPRZ00] and Lemma 2.2 of [CFR05b]) 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 (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 , for all suitable and . 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 -th moment of , we will estimate the probability that the local times in different points, say , belong to . In the subcritical regime (), 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 of the set of indices which orders the vertices so that we have the following: the walk first hits , accumulates a big local time in , then hits accumulates a big local time in , etc. When the walk has visited it does not come back to the vertices . The local times can thus be treated as if they were independent.
At criticality (), 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 observation - contained in Lemma 4.2.3 and already present in Corollary 1.3 of [CFR05b] - 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 -th moment.
3 Dimension two
3.1 General framework and upper bound
We now describe the general setup for the theorem. Consider a non-oriented connected infinite graph without loops, not necessary planar, and a continuous time random walk on , not necessary a nearest neighbour walk. As before, we take a starting point and write for the graph distance. We will also write
or simply if there is no confusion. Let denote the law of the walk starting from and the associated expectation. We introduce the first exit time of and the local times:
Finally we will denote the Green function, i.e.:
Notation: For two real-valued sequences and and for some parameter , we will denote if
and we will denote if
We now do the following assumptions on the graph and on the walk :
To ensure the existence of the GFF, we will need to assume:
For all , ,
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 . We assume that and that for all there exists a subset with points such that
and where we control the Green function as follows:
There exists such that:
For all , for every and , we can find a subset which can be thought of as a circle of radius centered at :
Finally, we assume that the jumps are not unreasonable:
For all , and ,
where is the first exit time of .
We now briefly discuss the above assumptions. Note that we have assumed that all the bounds do not depend on the starting point . 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 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 by with a function which goes to zero quickly enough as goes to zero. For instance, any positive power of would do.
As confirmed by the theorem below, a sensible definition of -thick points is given by
Assuming the above assumptions we have the following two -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
Square lattice. We first start to prove (17). By translation invariance, we can assume . We consider the discrete time random associated and we are going to abusively write to denote the first time the discrete time walk exits . Take 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 for some by an union bound. Secondly,
Theorem 2.3.9 of [LL10] gives estimates on the heat kernel and in particular implies that there exists such that for all , . Hence
We obtain (17) by taking .
which is linked to the Green function by:
If is such that , then
where the lower bound (resp. upper bound) is satisfied by all (resp. ). (17) implying that , we are thus left to show that the elements such that do not contribute to the sum in the equation (22). Thanks to (17), we have
which goes to zero as goes to infinity. It concludes the square lattice part of the lemma.
Proof of the upper bound of Theorem 3.1.1.
Let and . For every we obtain by Markov inequality:
But for every , under , is an exponential variable with mean . Hence by (14a),
The upper bound for the convergence in probability follows. To show that
we observe that, taking in (23),
decays exponentially and so is summable. Moreover, if ,
Hence the Borel–Cantelli lemma implies that
This concludes the proof of the upper bound on . The upper bound on follows from
3.2 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 , define the first exit time of and the set of -thick points in the ball :
Suppose that for all starting point , for all and ,
with decaying slower than any polynomial, i.e. . Then for all ,
A similar but weaker statement appears in [DPRZ01] and [Ros05] where they assumed that was bounded away from . The idea is to decompose the walk on the ball into several walks on smaller balls to bootstrap the probability we are interested in.
First of all, let us remark that if decays slower than any polynomial, then so does . Consequently, we can assume without loss of generality that the sequences in the statement of the lemma are non increasing.
Fix and take large and much smaller than such that . Let us introduce the stopping times
Let . If , then all the walks