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Register a new userThe infinite valley for a recurrent random walk in random environment
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We consider a one-dimensional recurrent random walk in random environment (RWRE). We show that the - suitably centered - empirical distributions of the RWRE converge weakly to a certain limit law which describes the stationary distribution of a random walk in an infinite valley. The construction of the infinite valley goes back to Golosov. As a consequence, we show weak convergence for both the maximal local time and the self-intersection local time of the RWRE and also determine the exact constant in the almost sure upper limit of the maximal local time.
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We study one-dimensional nearest neighbour random walk in site-random environment. We establish precise (sharp) large deviations in the so-called ballistic regime, when the random walk drifts to the right with linear speed. In the sub-ballistic regime, when the speed is sublinear, we describe the precise probability of slowdown.
Let $xi(n, x)$ be the local time at $x$ for a recurrent one-dimensional random walk in random environment after $n$ steps, and consider the maximum $xi^*(n) = max_x xi(n,x)$. It is known that $limsup xi^*(n)/n$ is a positive constant a.s. We prove that $liminf_n (logloglog n)xi^*(n)/n$ is a positive constant a.s.; this answers a question of P. Revesz (1990). The proof is based on an analysis of the {em valleys /} in the environment, defined as the potential wells of record depth. In particular, we show that almost surely, at any time $n$ large enough, the random walker has spent almost all of its lifetime in the two deepest valleys of the environment it has encountered. We also prove a uniform exponential tail bound for the ratio of the expected total occupation time of a valley and the expected local time at its bottom.
A random walk in a sparse random environment is a model introduced by Matzavinos et al. [Electron. J. Probab. 21, paper no. 72: 2016] as a generalization of both a simple symmetric random walk and a classical random walk in a random environment. A random walk $(X_n)_{nin mathbb{N}cup{0}}$ in a sparse random environment $(S_k,lambda_k)_{kinmathbb{Z}}$ is a nearest neighbor random walk on $mathbb{Z}$ that jumps to the left or to the right with probability $1/2$ from every point of $mathbb{Z}setminus {ldots,S_{-1},S_0=0,S_1,ldots}$ and jumps to the right (left) with the random probability $lambda_{k+1}$ ($1-lambda_{k+1}$) from the point $S_k$, $kinmathbb{Z}$. Assuming that $(S_k-S_{k-1},lambda_k)_{kinmathbb{Z}}$ are independent copies of a random vector $(xi,lambda)in mathbb{N}times (0,1)$ and the mean $mathbb{E}xi$ is finite (moderate sparsity) we obtain stable limit laws for $X_n$, properly normalized and centered, as $ntoinfty$. While the case $xileq M$ a.s. for some deterministic $M>0$ (weak sparsity) was analyzed by Matzavinos et al., the case $mathbb{E} xi=infty$ (strong sparsity) will be analyzed in a forthcoming paper.
Deterministic walk in an excited random environment is a non-Markov integer-valued process $(X_n)_{n=0}^{infty}$, whose jump at time $n$ depends on the number of visits to the site $X_n$. The environment can be understood as stacks of cookies on each site of $mathbb Z$. Once all cookies are consumed at a given site, every subsequent visit will result in a walk taking a step according to the direction prescribed by the last consumed cookie. If each site has exactly one cookie, then the walk ends in a loop if it ever visits the same site twice. If the number of cookies per site is increased to two, the walk can visit a site infinitely many times and still not end in a loop. Nevertheless the moments of $X_n$ are sub-linear in $n$ and we establish monotonicity results on the environment that imply large deviations.
We derive properties of the rate function in Varadhans (annealed) large deviation principle for multidimensional, ballistic random walk in random environment, in a certain neighborhood of the zero set of the rate function. Our approach relates the LDP to that of regeneration times and distances. The analysis of the latter is possible due to the i.i.d. structure of regenerations.
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