Passage time of the frog model has a sublinear variance

In this paper, we show that the passage time in the frog model on $\mathbb{Z}^d$ with $d\geq 2$ has a sublinear variance. The proof is based on the method introduced in \cite{DHS} combining with tessellation arguments to estimate the martingale difference. We also apply this method to get the linearity of the lengths of optimal paths.


Introduction
Frog models are simple but well-known models in the study of the spread of infection. In these models, individuals (also called frogs) move on the integer lattice Z d , which have one of two states infected (active) and healthy (passive). We assume that at the beginning, there is only one infected frog at the origin, and there are healthy frogs at other sites of Z d . When a healthy frog encounters with an infected one, it becomes infected forever. While the healthy frogs do not move, the infected ones perform independent simple random walks once they get infected. The object we are interested in is the long time behavior of the infected individuals.
To the best of our knowledge, the first result on frog models is due to Tecls and Wormald [19], where they proved the recurrence of the model (more precisely, they showed that the origin is visited infinitely often a.s.). Since then, there are numerous results on the behavior of the model under various settings of initial configurations, mechanism of walks, or underlying graphs, see [1,3,5,10,11,12,13,14]. In particular, Popov and some authors study the phase transition of the recurrence vs transience for the model with Bernoulli initial configurations and for the model with drift, see [2,9,11,18]. Another interesting feature in the frog model is that it can be described in the first passage percolation contexts, which is explained below. In fact, Alves, Machado and Popov used this property to prove a shape theorem [1]. Moreover, the large deviation estimates for the passage time are derived in [7,16] recently.
The frog model can be defined formally as follows. Let d ≥ 2 and {(S x j ) j∈N , x ∈ Z d } be independent SRWs such that S x 0 = x for any x ∈ Z d . For x, y ∈ Z d , let t(x, y) = inf{j : S x j = y}. The first passage time from x to y is defined by t(x i−1 , x i ) : x = x 0 , . . . , x k = y for some k .
The quantity T (x, y) can be seen as the first time when the frog at y becomes infected assuming that the frog at x was the only infected one at the beginning. For the simplicity of notation, we write T (x) instead of T (0, x).
Furthermore, a shape theorem for the set of active frogs has been also proved, see [1,Theorem 1.1]. The convergence (1.2), which can be seen as a law of large numbers, implies that for any x ∈ Z d the passage time T (x) grows linearly in |x| 1 . A natural question is whether the standard central limit theorem hold for T (x). The first task is to understand the behavior of variance of T (x). In [16], the author proves some large deviation estimates for T (x), see in particular Lemma 2.2 below. As a consequence, one can show that Var(T (x)) = O(|x| 1 (1 + log |x| 1 ) 2A ), for some constant A, see Corollary 2.3. However, this result is not enough to answer the question on standard central limit theorem.
Our main result is to show that the passage time has sublinear variance and thus the standard central limit theorem is not true. Theorem 1.1. There exists a positive constant C = C(d), such that for any x ∈ Z d , The sublinearity of variance as in Theorem 1.1, which is also called the superconcentration [6], was first discovered in the classical first passage percolation by Benjamini, Kali and Schram [4]. Hence, this result is sometimes called BKS-inequality. Chatterjee found the connection of superconcentration with chaos and multiple valleys in the gaussian polymer and SK model. This relation is expected to hold in general models. Therefore, the superconcentration is not only an interesting property itself but also an important object to study the structure of optimal paths and the energy landscape.
The superconcentration has been proved for several models such as the classical first passage percolation and the gaussian polymer model. In these proofs, one usually has to estimate the martingale difference carefully, which needs the model-dependent arguments. In the frog model, the correlation between passage times is problematic for this kind of estimate. A key observation to pass this difficulty is that the passage times are locally-dependent. Indeed, by large deviation estimates (see Lemma 2.1), T (x, y) ≤ C|x − y| 1 for some C > 0 with very high probability. Thus T (x, y) mainly depends on SRWs (S z · ) with |z − x| 1 ≤ C|x − y| 1 . Therefore, if the two pairs (x, y) and (u, v) are far enough from each other, the passage times T (x, y) and T (u, v) are weakly dependent. From this observation, using tessellation arguments, we decompose the martingale difference to some groups of the independent passage times. After that, we apply the percolation estimate to get the desired bound. This approach seems to be useful for other problems. Indeed, we also prove the linearity of the length of optimal path by using a similar method. Given x, y ∈ Z d , let us denote by Ø(x, y) the set of all optimal paths from x to y. We simply write Ø(x) for Ø(0, x). For any path γ = (y i ) ℓ i=1 ⊂ Z d , we denote l(γ) = ℓ the number of vertices in this path, and call it the length of γ. We will prove that the length of optimal paths from 0 to x grows linearly in |x| 1 . Proposition 1.2. There exist positive constants ε, c and C, such that for any • For any n ≥ 1, we denote by B(n) = [−n, n] d .
• Given y = y i ∈ γ, we defineȳ = y i+1 the next point of y in γ with the convention that y ℓ = y ℓ .
• We write y ∼ȳ ∈ γ ifȳ is the next point of y in γ.
• For L ≥ 1, we write such that f (x) ≤ Cg(x) for any x. • We use C > 0 for a large constant and ε for a small constant. Note that they may change from line to line.

1.2.
Organization of this paper. The paper is organized as follows. In Section 2, we present some preliminary results including large deviation estimates on the passage time, a lemma to control the tail distribution of maximal weight of paths in site-percolation, the introduction and properties of entropy. In Sections 3 and 4, we prove the main theorem 1. There exist a positive integer number C 1 and a positive constant ε 1 , such that for any x, y ∈ Z d and k ≥ 0, Notice that in [1], the authors only prove Lemma 2.1 for the case k = 0. However, we can easily generalize their arguments to all k ≥ 1. We safely leave the proof of this lemma to the reader. It follows from Lemma 2.1 that there exists C > 0 such that for any x ∈ Z d , The following concentration inequality is derived in [16].
As a direct consequence of Lemmas 2.1 and 2.2, we have Proof. We take a positive constant C sufficiently large such that Lemma 2.1 and (2.1) hold. By using the fact E(X 2 ) = ∞ 0 2tP(X ≥ t)dt for any non-negative random variable X, we get The first term of the right hand side (2.2) can be bounded from above by By Lemma 2.2, the second term is bounded from above by Finally, by (2.1) and Lemma 2.1, the third term is bounded from above by Combining these estimates, we get the conclusion.
Lemma 2.4. There exists a positive constant ε 2 , such that for any x, y ∈ Z d , and M ≥ 1 Proof. If |x − y| 1 ≤ M 2/3 , then the result follows from Lemma 2.1. Assume that |x − y| 1 ≥ M 2/3 . Then a well-known estimate for the trajectory of random walk (see [17,Proposition 2.1.2]) shows that for some positive constants c and C, Therefore, for some c, C > 0.

2.2.
A result on the maximal weight of paths in site-percolation. LetP L be the set of self-avoiding nearest-neighbor paths in B(L) whose length is bounded by L, i.e., Let {X x } x∈Z d be a collection of independent and identical distribution random variables such that P(X x = 1) = 1 − P(X x = 0) = p with a parameter p ∈ [0, 1]. For any path γ, we define X(γ) = x∈γ X x the weight of γ. The maximal weight of paths inP L and P L are defined respectively asX Note that for any γ ∈ P L , there existsγ ∈P L such that γ ⊂γ. This implies X L ≤X L .
The tail distribution and expectation ofX L can be controlled as in the following lemma.
In particular, the above estimates hold if we replaceX L by X L .
We notice that in [8], the authors prove these results for the edge-percolation, i.e. for the setting where (X e ) e∈E d (with E d the edge set of Z d ) are the edge-indexed i.i.d. Bernoulli random variables and Q L is the set of edge-paths in B(L). However, their proof can be easily adapted to the case of site-percolation as in Lemma 2.5. We also remark that in Lemma 6.8 of [8], the authors only stated Part (ii), but in fact, they have proved (i) and derived (ii) from (i).

Entropy.
We first recall the definition of entropy with respect to a probability measure. Let (Ω, F , µ) be a probability space and X ∈ L 1 (Ω, µ) be a non-negative. Then Note that by Jensen's inequality, Ent µ (X) ≥ 0. The following tensorization property of entropy is proved in [8].
where Ent i (X) is the entropy of X(ω) = X((ω 1 , . . . , ω i , . . .)) with respect to µ i , as a function of the i-th coordinate (with all other values fixed).
In the following lemma, we prove a generalization of Bonami inequality for simple random variables.
. . , k} → R be a function and ν be the uniform distribution on {1, . . . , k}. Then where E is the expectation with respect to two independent random variables U,Ũ , which have the same distribution ν.
Proof. Let us denote a i = f (i). Then where we have used Jensen's inequality in the last inequatliy. Moreover, On the other hand, Hence, which proves Lemma 2.7.
3. Proof of Theorem 1.1 3.1. Spatial average of the passage time. We consider a spatial average of T (x) defined by Proof. For any variables X and Y , by writingX = X − E(X) and ||X|| 2 = (E(X 2 )) 1/2 and using Cauchy-Schwartz inequality, we get We aim to apply (3.1) for T (x) and F m . Observe that by translation invariance. By Corollary 2.3, Using the subadditivity (1.1), Using Cauchy-Shwartz inequality and the translation invariance, this is further bounded from above by Using Lemma 2.1 and the union bound, we have Therefore, by a similar argument as in Corollary 2.3, we have Combining (3.1)-(3.4), we get the desired result.

3.2.
Martingale decomposition of F m and the proof of Theorem 1.1. Enumerate the vertices of Z d as x 1 , x 2 , . . .. We consider the martingale decomposition of F m as follows with F k the sigma-algebra generated by SRWs {(S xi j ) j∈N , i = 1, . . . , k} and F 0 the trivial sigmaalgebra. In [8], using Falik-Samorodnitsky lemma, the authors give an upper bound for variance of F m in term of Ent(∆ 2 k ), and E(|∆ k |).
Now, our main task is to estimate Ent(∆ 2 k ) and E(|∆ k |). Proposition 3.3. As |x| 1 tends to infinity, . Therefore, using Propositions 3.1, 3.3 and Lemma 3.2, for any ε > 0, there exists a positive constant C, such that We precise the dependence of passage time on trajectories of SRWs by writing For any k, let us define Then X k (u, v) is a function of trajectories of (S xi . ) i≤k , so we write Let (S x . ) x∈Z d be an independent copy of (S x . ) x∈Z d . We observe that , and E <k , E k , andẼ k denote the expectations with respect to SRWs (S xi . ) i<k , (S x k . ) and (S x k . ) respectively. Then the inequality (3.7) becomes By symmetry, For any u, v ∈ Z d , we choose an optimal path for T (u, v) with a deterministic rule breaking ties and denote it by γ u,v . We observe that if withx k the next point of x k in γ u,v (recall also that we denote by y ∼ȳ ∈ γ ifȳ is the next point of y in γ). Due to the subadditivity, It is clear that the optimal path for T (u, x k ) does not use the simple random walk (S x k · ). Hence, In addition, sincex k is the next point of x k in γ u,v , the optimal path for T (x k , v) does not use the simple random walk (S x k · ). Thus It follows from (3.10)-(3.13) thatT Therefore, we have Combining (3.6), (3.8), (3.9) and (3.14), we get where E ⊗2 is the expectation with respect to two independent collections of SRWs (S xi . ) i∈N and (S xi . ) i∈N and let Notice that for the second equation, we have used the invariant translation. Let us define as the passage time from u to v not using the frog at z, and set Then, it holds that Using (3.16), we obtain where we have used the Cauchy-Schwartz inequality in the second inequality.
These yield that We postpone the proof of this lemma for a while.
Then, for any L ≥ m = |x| By the union bound, Lemma 2.1 and Lemma 2.4, we have for some constant ε > 0.
Combining this inequality with (3.15), (3.17), (3.18) and Lemma 3.4, we obtain that there exists C > 0 such that for any k ≥ 1 Since T (x) ≥ |γ 0,x | 1 , by using Lemma 2.1, for any L ≥ C 1 |x| 1 We now turn to prove Proposition 3.3 (i). To estimate Ent(∆ k ), we decompose the simple random walks (S xi . ) into the sum of i.i.d. random variables. More precisely, for any x i ∈ Z d and j ≥ 1, we write where (ω i,r ) i,r≥1 is an array of i.i.d. uniform random variables taking value in the set of canonical coordinates in Z d , denoted by Therefore, we can view T (u, v) and F m as a function of (ω i,r ), and hence we sometimes write T (u, v) = T (u, v, ω) to precise the dependence of T (u, v) on ω. We define where Ω i,j is a copy of B d . The measure on Ω is π = i,j∈N π i,j , where π i,j is the uniform measure on Ω i,j . Then we can consider F m as a random variable on the probability space (Ω, π). Given ω ∈ Ω, e ∈ B d and i, j ∈ N, we define a new configuration ω i,j,e as We define where the expectation runs over two independent random variables U andŨ , with the same law as the uniform distribution on B d .
By Lemma 2.7, Now using the same arguments as in Lemma 6.3 in [8], we can show that for any i, j. Combining this equation with (3.26), we get the desired result.

Proof of Proposition 3.3 (i). Using Lemma 3.6 and the Cauchy-Schwartz inequality, we get
On the other hand, We observe that if x i ∈ γ z,z+x , or x i ∈ γ z,z+x but T (x i ,x i ) < j, then Otherwise, assume that x i ∈ γ z,z+x and T (x i ,x i ) ≥ j. Then for any e ∈ B d , since if we only replace ω i,j by e, after t(x i ,x i ) (also equals to T (x i ,x i ), as x i ∼x i ∈ γ 0,x ) steps, the simple random walk (S xi . ) arrives atx i − e + ω i,j . Moreover, Hence, we reach Furthermore, since ω differs from ω i,j,U only in the trajectory of (S xi . ), for any u, v ∈ Z d , Therefore, we have

This yields that
Now using the same arguments for (3.18) and (3.20), we get Lemma 3.7. As |x| 1 tends to infinity, Lemma 3.8. There exists a positive constant C, such that for any L ≥ 1, We postpone the proofs of the above two lemmas for a while and first complete the proof of Proposition 3.3. Combining (3.23), (3.29), (3.30) and Lemmas 3.7 and 3.8, we get This estimate holds with all z ∈ B(m), so we can conclude the proof of Proposition 3.3 by using (3.27) and Lemma 3.6.

Tessellation estimates.
In this section, we will prove Lemmas 3.4, 3.7 and 3.8. We first observe that the simple union bound is not sharp enough to prove the lemmas and that the main difficulty comes from the correlations of passage times. To overcome this, we establish new techniques combining tessellation arguments and percolation estimates (Lemma 2.5).
Proof of Lemma 3.7. For any γ = (y i ) ℓ i=1 , we define Then, we can express By a similar argument as in Lemma 2.2, the second term can be bounded from above by which goes to 0 as |x| 1 → ∞.
To estimate the first term, under the condition T (x) ≤ C|x| 1 , we will show that for any M ≥ 1, , with some constants ε > 0 and C > 0. Then it follows from (3.32) that the first term can be bounded from above by which proves Lemma 3.7.
Now it remains to prove (3.34). The general idea is to cover Z d by groups of boxes such that in each group the numbers of two consecutive points in the optimal path having distance M in different boxes are dominated by independent random variables. Then we will apply Lemma 2.5 to get the desired estimate.
Lemma 3.9. The groups of boxes that we have constructed above satisfy (c) and (d).
Proof. The condition (d) is trivial by construction. We will prove that (c) holds. Assume that u, v ∈ Z d and |u−v| 1 ≤ M . We consider Then by (a) and (c), Notice that if y ∼ȳ ∈ γ 0,x , then T (y,ȳ) = t(y,ȳ). Thus for any (i, z), M,i,z = ∅}. Since we assume that T (x) ≤ C 1 |x| 1 , we have γ 0,x ∈ P C1|x|1 , which implies Combining this inequality with (3.36) and (3.37) yields that  Before going into the rest of the proofs, we will prove the same estimates as in Lemma 2.1 for T 1 and T 2 . By repeating the arguments of the proof of Lemma 2.1 ([1, Lemma 4.2]), we can show that there exist positive constants C and ε, such that for any y, z ∈ Z d , and k ≥ C|y| 1 , By the union bound, for k ≥ C 2 |y| 1 with C 2 = 2C, we have We observe also that if T (y) ≤ k then T [z] (0, y) = T (y) for z ∈ B(k). Therefore, for k ≥ C 3 |y| 1 with C 3 = max{C 1 , C 2 }, with some ε 3 > 0. From now on, for simplicity of notation we use C 1 for all C 1 , C 2 , C 3 , and ε 1 for all ε 1 , ε 2 , ε 3 . It means that for k ≥ C 1 |y| 1 , Proof of Lemma 3.4. We begin with Part (ii), which is easier than (i). Observe that Using the union bound and (3.46), for any k ≥ 2dC 1 L, The last two inequalities yield that We now prove (i). For any γ = (y i ) ℓ i=1 ∈ P L , we definē We shall apply the same arguments as in the proof of Lemma 3.7 to deal with the sum above. For each M, k we tessellate Z d to groups of boxes whose size equals 2(C 1 M + k). Using analogous arguments to prove (3.41) and (3.42), we can show that Note that if T (x) ≤ C 1 |x| 1 , then γ ∈ P C1|x|1 for any γ ∈ Ø(x), and thus Therefore, 2M d+2 x ≤ e −|x| ε 1 , (4.8) for some ε > 0. We now only need to take K large enough such that C d /K < 1/2 and s M ≥ A 1 for any M ≥ K. Then by (4.4), (4.5), (4.6) and (4.8), if we take c > 0 sufficiently small so that 1 − cK ≥ C d /K, we get P min γ∈Ø(x) l(γ) < c|x| 1 ≤ P(T (x) > C 1 |x| 1 ) + P(E c ) + P