Central limit theorem for the excited random walk in dimension $d\geq 2$

An excited random walk with bias parameter $p\in(1/2,1]$ is a discrete time nearest neighbor random walk $(X_{n})_{n\geq 0}$ on the lattice $\mathbb{Z}^{d}$ obeying the following rule: when at time $n$ the walk is at a site it has already visited before time $n$, it jumps uniformly at random to one of the $2d$ neighboring sites. On the other hand, when the walk is at a site it has not visited before time $n$, it jumps with probability $p/d$ to the right, probability $(1-p)/d$ to the left, and probability $1/(2d)$ to the other nearest neighbor sites.

The excited random walk was introduced in 2003 by Benjamini and Wilson [1], motivated by previous works of [5, 4] and [10] on self-interacting Brownian motions. Variations on this model have also been introduced. The excited random walk on a tree was studied by Volkov [15]. The so called multi-excited random walk, where the walk gets pushed towards a specific direction upon its first $M_{x}$ visits to a site $x$, with $M_{x}$ possibly being random, was introduced by Zerner in [16] (see also [17] and [9]).

In [1], Benjamini and Wilson proved that for every value of $p\in(1/2,1]$ and $d\geq 2$, excited random walks are transient. Furthermore, they proved that for $d\geq 4$,

where $(e_{i}:1\leq i\leq d)$ denote the canonical generators of the group $\mathbb{Z}^{d}$. Subsequently, Kozma extended (1) in [7] and [8] to dimensions $d=3$ and $d=2$. Then, in [14], relying on the lace expansion technique, van der Hofstad and Holmes proved that a weak law of large numbers holds when $d>5$ and $p$ is close enough (depending on $d$) to $1/2$, and that a central limit theorem hold when $d>8$ and and $p$ is close enough (depending on $d$) to $1/2$.

In this paper, we prove that the biased coordinate of the excited random walk satisfies a law of large numbers and a central limit theorem for every $d\geq 2$ and $p\in(1/2,1]$.

Our proof is based on the well-known construction of regeneration times for the random walk, the key issue being to obtain good tail estimates for these regeneration times. Indeed, using estimates for the so-called tan points of the simple random walk, introduced in [1] and subsequently used in [7, 8], it is possible to prove that, when $d\geq 2$, the number of distinct points visited by the excited random walk after $n$ steps is, with large probability, of order $n^{3/4}$ at least. Since the excited random walk performs a biased random step at each time it visits a site it has not previously visited, the $e_{1}$-coordinate of the walk should typically be at least of order $n^{3/4}$ after $n$ steps. Since this number is $o(n)$, this estimate is not good enough to provide a direct proof that the walk has linear speed. However, such an estimate is sufficient to prove that, while performing $n$ steps, the walk must have many independent opportunities to perform a regeneration. A tail estimate on the regeneration times follows, and in turn, this yields the law of large numbers and the central limit theorem, allowing for a full use of the spatial homogeneity properties of the model. When $d\geq 3$, it is possible to replace, in our argument, estimates on the number of tan points by estimates on the number of distinct points visited by the projection of the random walk on the $(e_{2},\ldots,e_{d})$ coordinates – which is essentially a simple random walk on $\mathbb{Z}^{d-1}$. Such an observation was used in [1] to prove that (1) holds when $d\geq 4$. Plugging the estimates of [6] in our argument, we can rederive the law of large numbers and the central limit theorem when $d\geq 4$ without considering tan points. Furthermore, a translation of the results in [2] and [11] about the volume of the Wiener sausage to the random walk situation considered here, would allow us to rederive our results when $d=3$, and to improve the tail estimates for any $d\geq 3$.

The regeneration time methods used to prove Theorem 1 could also be used to describe the asymptotic behavior of the configuration of the vertices as seen from the excited random walk. Let $\Xi:=\{0,1\}^{\mathbb{Z}^{d}\setminus\{0\}}$, equipped with the product topology and $\sigma-$algebra. For each time $n$ and site $x\neq X_{n}$, define $\beta(x,n):=1$ if the site $x$ was visited before time $n$ by the random walk, while $\beta(x,n):=0$ otherwise. Let $\zeta(x,n):=\beta(x-X_{n},n)$ and define

We call the process $(\zeta(n))_{n\in\mathbb{N}}$ the environment seen from the excited random walk. It is then possible to show that if $\rho(n)$ is the law of $\zeta(n)$, there exists a probability measure $\rho$ defined on $\Xi$ such that

weakly.

In the following section of the paper we introduce the basic notation that will be used throughout. In Section 3, we define the regeneration times and formulate the key facts satisfied by them. In Section 4 we obtain the tail estimates for the regeneration times via a good control on the number of tan points. Finally, in Section 5, we present the results of numerical simulations in dimension $d=2$ which suggest that, as a function of the bias parameter $p$, the speed $v(p,2)$ is an increasing convex function of $p$, whereas the variance $\sigma(p,2)$ is a concave function which attains its maximum at some point strictly between $1/2$ and $1$.

Let $\mathbf{b}:=\{e_{1},\ldots,e_{d},-e_{1},\ldots,-e_{d}\}$. Let $\mu$ be the distribution on $\mathbf{b}$ defined by $\mu(+e_{1})=p/d$, $\mu(-e_{1})=(1-p)/d$, $\mu(\pm e_{j})=1/2d$ for $j\neq 1$. Let $\nu$ be the uniform distribution on $\mathbf{b}$. Let $\mathcal{S}_{0}$ denote the sample space of the trajectories of the excited random walk starting at the origin:

For all $k\geq 0$, let $X_{k}$ denote the coordinate map defined on $\mathcal{S}_{0}$ by $X_{k}((z_{i})_{i\geq 0}):=z_{k}$. We will sometimes use the notation $X$ to denote the sequence $(X_{k})_{k\geq 0}$. We let $\mathcal{F}$ be the $\sigma-$algebra on $\mathcal{S}_{0}$ generated by the maps $(X_{k})_{k\geq 0}$. For $k\in\mathbb{N}$, the sub-$\sigma-$algebra of $\mathcal{F}$ generated by $X_{0},\ldots,X_{k}$ is denoted by $\mathcal{F}_{k}$. And we let $\theta_{k}$ denote the transformation on $\mathcal{S}_{0}$ defined by $(z_{i})_{i\geq 0}\mapsto(z_{k+i}-z_{k})_{i\geq 0}$. For the sake of definiteness, we let $\theta_{+\infty}((z_{i})_{i\geq 0}):=(z_{i})_{i\geq 0}$. For all $n\geq 0$, define the following two random variables on $(\mathcal{S}_{0},\mathcal{F})$:

(Note that, with this definition, $J_{0}=1$.)

We now call $\mathbb{P}_{0}$ the law of the excited random walk, which is formally defined as the unique probability measure on $(\mathcal{S}_{0},\mathcal{F})$ satisfying the following conditions: for every $k\geq 0$,

• •

  on $X_{k}\notin\{X_{i};\,0\leq i\leq k-1\}$, the conditional distribution of $X_{k+1}-X_{k}$ with respect to $\mathcal{F}_{k}$ is $\mu$;

• •

  on $X_{k}\in\{X_{i};\,0\leq i\leq k-1\}$, the conditional distribution of $X_{k+1}-X_{k}$ with respect to $\mathcal{F}_{k}$ is $\nu$.

We now define the regeneration times for the excited random walk (see [13] for the same definition in the context of random walks in random environment). Define on $(\mathcal{S}_{0},\mathcal{F})$ the following $(\mathcal{F}_{k})_{k\geq 0}$-stopping times: $T(h):=\inf\{k\geq 1;\,X_{k}\cdot e_{1}>h\}$, and $D:=\inf\{k\geq 1;\,X_{k}\cdot e_{1}=0\}$. Then define recursively the sequences $(S_{i})_{i\geq 0}$ and $(D_{i})_{i\geq 0}$ as follows: $S_{0}:=T(0)$, $D_{0}:=S_{0}+D\circ\theta_{S_{0}}$, and $S_{i+1}:=T(r_{D_{i}})$, $D_{i+1}:=S_{i+1}+D\circ\theta_{S_{i+1}}$ for $i\geq 0$, with the convention that $S_{i+1}=+\infty$ if $D_{i}=+\infty$, and, similarly, $D_{i+1}=+\infty$ if $S_{i+1}=+\infty$. Then define $K:=\inf\{i\geq 0;\,D_{i}=+\infty\}$ and $\kappa:=S_{K}$ (with the convention that $\kappa=+\infty$ when $K=+\infty$).

The key estimate for proving our results is stated in the following proposition.

A consequence of the above proposition is that, under $\mathbb{P}_{0}$, $\kappa$ has finite moments of all orders, and also $X_{\kappa}$, since the walk performs nearest-neighbor steps. We postpone the proof of Proposition 1 to Section 4.

Now define the sequence of regeneration times $(\kappa_{n})_{n\geq 1}$ by $\kappa_{1}:=\kappa$ and $\kappa_{n+1}:=\kappa_{n}+\kappa\circ\theta_{\kappa_{n}}$, with the convention that $\kappa_{n+1}=+\infty$ if $\kappa_{n}=+\infty$. For all $n\geq 0$, we denote by $\mathcal{F}_{\kappa_{n}}$ the completion with respect to $\mathbb{P}_{0}-$negligible sets of the $\sigma-$algebra generated by the events of the form $\{\kappa_{n}=t\}\cap A$, for all $t\in\mathbb{N}$, and $A\in\mathcal{F}_{t}$.

The following two propositions are analogous respectively to Theorem 1.4 and Corollary 1.5 of [13]. Given Lemma 1 and Lemma 2, the proofs are completely similar to those presented in [13], noting that the process $(\beta(n),X_{n})_{n\in\mathbb{N}}$ is strongly Markov, so we omit them, and refer the reader to [13].

For future reference, we state the following result.

A consequence of Proposition 1 is that $\mathbb{E}_{0}(\kappa|D=+\infty)<+\infty$ and $\mathbb{E}_{0}(|X_{\kappa}||D=+\infty)<+\infty$. Since $\mathbb{P}_{0}(\kappa\geq 1)=1$ and $\mathbb{P}_{0}(X_{\kappa}\cdot e_{1}\geq 1)=1$, $\mathbb{E}_{0}(\kappa|D=+\infty)>0$ and $\mathbb{E}_{0}(X_{\kappa}\cdot e_{1}|D=+\infty)>0$. Letting $v(p,d):=\frac{\mathbb{E}_{0}(X_{\kappa}\cdot e_{1}|D=+\infty)}{\mathbb{E}_{0}(\kappa|D=+\infty)}$, we see that $0<v(p,d)<+\infty$.

The following law of large numbers can then be proved, using Proposition 3, exactly as Proposition 2.1 in [13], to which we refer for the proof.

Another consequence of Proposition 1 is that $\mathbb{E}_{0}(\kappa^{2}|D=+\infty)<+\infty$ and $\mathbb{E}_{0}(|X_{\kappa}|^{2}|D=+\infty)<+\infty$. Letting $\sigma^{2}(p,d):=\frac{\mathbb{E}_{0}(\left[X_{\kappa}\cdot e_{1}-v(p,d)\kappa\right]^{2}|D=+\infty)}{\mathbb{E}_{0}(\kappa|D=+\infty)}$, we see that $\sigma(p,d)<+\infty$. That $\sigma(p,d)>0$ is explained in Remark 1 below.

The following functional central limit theorem can then be proved, using Proposition 3, exactly as Theorem 4.1 in [12], to which we refer for the proof.