Maximum and minimum of local times for two-dimensional random walk

We obtain the leading orders of the maximum and the minimum of local times for the simple random walk on the two-dimensional torus at time proportional to the cover time. We also estimate the number of points with large (or small) values of the local times. These are analogues of estimates on the two-dimensional Gaussian free fields by Bolthausen, Deuschel, and Giacomin [Ann. Probab., 29 (2001)] and Daviaud [Ann. Probab., 34 (2006)], but we have different exponents from the case of the Gaussian free field.


Introduction
The theory of local times of random walks is very profound.It is well-known that local times of random walks have close relationships with the Gaussian free field(GFF).The connection goes back to [9].Eisenbaum, Kaspi, Marcus, Rosen, and Shi [10] gave a powerful equivalence in law called "generalized second Ray-Knight theorem" (see Remark 1.1).Using the theorem, Ding, Lee, and Peres [5] established a useful connection between the expected maximum of the GFF and the cover time, and quite recently Zhai [17] strengthened the result by constructing a coupling of the occupation time filed and the GFF (see Theorem 2.6).
Much efforts have been made to study local times of the simple random walk on Z 2 .Erdős and Taylor [11] obtained an estimate on the maximum of local times of the simple random walk on Z 2 by time n.Dembo, Peres, Rosen and Zeitouni [6] improved the result; they gave the leading order of the maximum and estimated the number of "favorite points" (see also [15]).Okada [14] obtained a corresponding estimate on frequently visited sites in the inner boundary of the random walk range.Sznitman [16] studied convergences of occupation time fields and related the fields to the GFF.
As mentioned above, works [11,6,15] are closely linked to the study of extremes of the two-dimensional GFF.Bolthausen, Deuschel, and Giacomin [2] obtained the leading order of its maximum (see Remark 1.4).Daviaud [3] estimated the number of points with large values of the GFF (see Remark 1.4).
In this paper, we study the maximum and the minimum of local times of the simple random walk on the two-dimensional torus at time proportional to the cover time.While similar work has been done in [2,3] for the GFF, one cannot apply their results to deduce corresponding local time estimates, and indeed considerable amounts of efforts are needed to obtain such estimates.We also note that the exponents for the local times are different from those of the GFF (see Theorem 1.2 and Remark 1.4).
To state our results, we begin with some notation.We will write Z 2 N to denote the two-dimensional discrete torus with N 2 vertices.Let X = (X t ) t≥0 be the continuoustime simple random walk on Z 2 N with exponential holding times of parameter 1.Let P x be the law of X starting from x ∈ Z 2 N .We define the local time of X by and the inverse local time by τ t := inf{s ≥ 0 : L N s (0) > t}, t ≥ 0. We will take the following time parameter 2 and that 4 π N 2 (log N) 2 is close to the cover time of Z 2 N (see Lemma 2.4 and Theorem 1.1 in [7]).We define sets of "thick points" and "thin points" by We will say that N with a measure P. The generalized second Ray-Knight theorem [10] says that for all t ≥ 0, under the measure P 0 × P, In particular, fixing N, we have By (1.4), one can expect that (1.1) and (1.2) will be close in law to corresponding level sets of the GFF (but not exactly, as we see in Theorem 1.2 and Remark 1.4).We note that one cannot deduce local time estimates corresponding to [2,3] from (1.3) or (1.4).

Corollary 1.3 (i)
For all θ > 0 and ε > 0, the following holds with high probability (under P 0 ): (ii) For all θ > 1 and ε > 0, the following holds with high probability (under P 0 ): x∈V N be the GFF on V N with zero boundary conditions.Bolthausen, Deuschel, and Giacomin [2] obtained the leading order of max x∈V N hN x : for all ε > 0, Daviaud [3] showed that the following holds with high probability for all ε > 0 and η ∈ (0, 1): We note that one can obtain an estimate similar to (1.5) for the GFF with periodic boundary conditions by using Theorem 1.2 and Theorem 2.5, 2.6 below.
Remark 1.5 As mentioned before, t 1 is close to the cover time for the walk X (see Theorem 2.5 below).Thus, it is clear that for θ ∈ (0, 1), we have In order to give an intuitive explanation of the exponent in Theorem 1.2(i), let us give additional notation.Let We define its boundary by N \A, d(x, y) = 1 for some x ∈ A}, and the hitting time of A by T A := inf{t ≥ 0 : X t ∈ A}.We will write T x to denote We define a sequence of stopping times as follows: where θ t ,t ≥ 0 is the shift operator.We define local times of excursions as follows: , j ≥ 1.
We now give heuristics about the exponent in Theorem 1.2(i).Let K n := n b e n n 3n , where b is a positive constant.We will consider the simple random walk on K n and 0 ≤ ℓ ≤ n − 1, we write N x ℓ to denote the number of excursions from ∂ D(x, r n,ℓ+1 ) to ∂ D(x, r n,ℓ ) up to time τ t θ .By concentration estimates (see Lemma 2.2 and 2.4), Thus, we have By the law of large numbers, if where y is a fixed point in ∂ D(x, r n,n ), and we have used an estimate on Green's functions (see Lemma 2.1).Hence, we have To obtain the order of |L + K n (η, θ )|, we should estimate the probability ) ) ≈ 1 2 (see Lemma 2.1), we can reduce the problem to the case of the simple random walk on {0, . . ., n}; we need to know the probability of the event that the walk traverses 6(θ + 2η √ θ )n 2 log n times from n to n − 1 until it crosses N x 0 times from 1 to 0. By this observation, (1.6) and a large deviation estimate, we have Therefore, if |L + K n (η, θ )| is concentrated around its expectation, we have The organization of the paper is the following.Section 2 gives preliminary lemmas.In Section 3, we prove Theorem 1.2(i).The proof is based on the"refined second moment method" in [8,15].In Section 4, we prove Theorem 1.2(ii).We will write c 1 , c 2 , . . . to denote positive universal constants whose values are fixed within each argument.We use c 1 (θ ), c 2 (θ ) . . .for positive constants which depend only on θ .Given a sequence (ε N ) N≥0 , we will write ε

Preliminary lemmas
In this section, we collect lemmas which are useful in the proof of Theorem 1.2.We will use the following basic estimates on the two-dimensional random walk.See, for example, Theorem 1.6.6,Proposition 1.6.7,and Exercise 1.6.8 in [13].

Lemma 2.1 (i)
There exist c 1 , c 2 > 0 such that the following hold for all 0 < R < N 2 , x ∈ Z 2 N , and x 0 ∈ D(x, R): The following lemma relates time to the number of excursions.
Lemma 2.2 There exist c 1 , c 2 , c 3 such that the following holds for all r, R with , and M ∈ N: Proof.The proof is almost the same as that of Lemma 3.2 in [8] since Lemma 3.1 of [8] holds even for the continuous-time simple random walk.We will use the following moment estimate on local times.
The following is a special version of Lemma 2.1 in [4].
Lemma 2.4 There exists c 1 > 0 such that for all t > 0 and λ ≥ 1, Proof.Note that the definition of the inverse local time in [4] is slightly different from ours; it corresponds to τ 4t in our notation.Since the effective resistances between vertices in Z 2 N are of order log N, the statement follows from Lemma 2.1 of [4].The following theorem is about the number of "late points" of X.
Theorem 2.5 For all ε > 0 and η ∈ (0, 1), the following holds with high probability (under P 0 ): Furthermore, for all η > 1, Proof.Recall that the holding times of X are independent exponential variables with mean 1.Thus, it is clear that Theorem 2.5 follows immediately from Proposition 1.1 in [8], Theorem 1.1 in [7], and the law of large numbers for the variables.
The following theorem connects "thick points", "thin points" and the GFF.

Proof of Theorem 1.2(i)
Given Theorem 2.6, the upper bound of Theorem 1.2(i) is easy.
Proof of the upper bound of Theorem 1.2(i).Fix θ > 0, ε > 0, and η > 0. Let (h N x ) x∈Z 2 N be the GFF on Z 2 N .We have for all λ > 0, where we have used the symmetry of the GFF in (3.2) and Theorem 2.6 in (3.1) and From now on, we prove the lower bound of Theorem 1.2(i) by applying the methods in [8,15].First, we define the notion that a point is "successful".Set where γ ∈ [b, b + 4] and b is a sufficiently large positive constant.Since K n 's take values over all sufficiently large positive integers, we may only consider the subsequence.
From now, we will consider the simple random walk on Z 2 K n .Given η ∈ (0, 1 + 1 2 √ θ ), we set up to random time We will say that x is successful if

Remark 3.2 We give an intuition about Definition 3.1. Assume that
We already know that under this assumption, N x 0 ≈ 6θ n 2 log n and 0≤ℓ≤n−1 behaves like a linear function.

6) and (1.7)). Due to a recent work by Belius
and Kistler [1], one expects that conditioned on Therefore, we see that ( N x ℓ ) 0≤ℓ≤n−1 would typically look like a linear function in ℓ with N x 0 ≈ √ n 0 and N x n−1 ≈ √ n n−1 (see Figure 1).We used this insight in Definition 3.1.Note that our framework is quite different from those in [8,15] and so is the definition of "successful".
We have . Taking b large enough, by Proposition 3.4 and 3.5, we have Thus, we have By (3.6), Proposition 3.4, and the Paley-Zygmund inequality, the following holds with high probability: The lower bound of Theorem 1.2(i) follows from (3.7) and Proposition 3.3.
For the rest of this section, we will prove Proposition 3.3-3.5.Proof of Proposition 3.3.We will prove the following: and x is successful for some x ∈ Z 2 K n \D(0, r n,0 )] → 0 as n → ∞. ( The statement in Proposition 3.3 follows immediately from this.The probability in (3.8) is bounded above by I 1 + I 2 + I 3 , where ) where λ n := (1+1/n 1/4 )2/π(K n ) 2 log(r n,0 /r n,1 . By Lemma 2.2 and 2.4, we have From now, we will prove K n \D(0, r n,0 ).Set By Lemma 2.1(i) and 2.3 together with the Chebyshev inequality and the strong Markov property, we have for ϕ > 0, where Taking ϕ at which f n (ϕ) attains the maximum, we have Therefore, we have proved Proof of Proposition 3.4.Fix x ∈ Z 2 K n \D(0, r n,0 ).By Lemma 2.1(ii) and the strong Markov property, we have where where f (u) := (1 + u) log(1 + u) − u logu − (1 + u) log2, u > 0 and we have used the Taylor expansion of f around 1 in (3.14).Therefore, we have by (3.13) and (3.15) By a similar argument, we can obtain the upper bound of q n .

Proof of Theorem 1.2(ii)
In this section, we prove Theorem 1.2(ii).First, we show the lower bound.