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Register a new userQuenched tail estimate for the random walk in random scenery and in random layered conductance II
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This is a continuation of our earlier work [Stochastic Processes and their Applications, 129(1), pp.102--128, 2019] on the random walk in random scenery and in random layered conductance. We complete the picture of upper deviation of the random walk in random scenery, and also prove a bound on lower deviation probability. Based on these results, we determine asymptotics of the return probability, a certain moderate deviation probability, and the Green function of the random walk in random layered conductance.
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We discuss the quenched tail estimates for the random walk in random scenery. The random walk is the symmetric nearest neighbor walk and the random scenery is assumed to be independent and identically distributed, non-negative, and has a power law tail. We identify the long time aymptotics of the upper deviation probability of the random walk in quenched random scenery, depending on the tail of scenery distribution and the amount of the deviation. The result is in turn applied to the tail estimates for a random walk in random conductance which has a layered structure.
In this paper, we are interested in the asymptotic behaviour of the sequence of processes $(W_n(s,t))_{s,tin[0,1]}$ with begin{equation*} W_n(s,t):=sum_{k=1}^{lfloor ntrfloor}big(1_{{xi_{S_k}leq s}}-sbig) end{equation*} where $(xi_x, xinmathbb{Z}^d)$ is a sequence of independent random variables uniformly distributed on $[0,1]$ and $(S_n)_{ninmathbb N}$ is a random walk evolving in $mathbb{Z}^d$, independent of the $xi$s. In Wendler (2016), the case where $(S_n)_{ninmathbb N}$ is a recurrent random walk in $mathbb{Z}$ such that $(n^{-frac 1alpha}S_n)_{ngeq 1}$ converges in distribution to a stable distribution of index $alpha$, with $alphain(1,2]$, has been investigated. Here, we consider the cases where $(S_n)_{ninmathbb N}$ is either: a) a transient random walk in $mathbb{Z}^d$, b) a recurrent random walk in $mathbb{Z}^d$ such that $(n^{-frac 1d}S_n)_{ngeq 1}$ converges in distribution to a stable distribution of index $din{1,2}$.
We consider a nearest-neighbor, one-dimensional random walk ${X_n}_{ngeq 0}$ in a random i.i.d. environment, in the regime where the walk is transient with speed v_P > 0 and there exists an $sin(1,2)$ such that the annealed law of $n^{-1/s} (X_n - n v_P)$ converges to a stable law of parameter s. Under the quenched law (i.e., conditioned on the environment), we show that no limit laws are possible. In particular we show that there exist sequences {t_k} and {t_k} depending on the environment only, such that a quenched central limit theorem holds along the subsequence t_k, but the quenched limiting distribution along the subsequence t_k is a centered reverse exponential distribution. This complements the results of a recent paper of Peterson and Zeitouni (arXiv:0704.1778v1 [math.PR]) which handled the case when the parameter $sin(0,1)$.
In this paper we consider a d-dimensional scenery seen along a simple symmetric branching random walk, where at each time each particle gives the color record it is seeing. We show that we can a.s. reconstruct the scenery up to equivalence from the color record of all the particles. For this we assume that the scenery has at least 2d + 1 colors which are i.i.d. with uniform probability. This is an improvement in comparison to [22] where the particles needed to see at each time a window around their current position. In [11] the reconstruction is done for d = 2 with only one particle instead of a branching random walk, but millions of colors are necessary.
We consider random walk on dynamical percolation on the discrete torus $mathbb{Z}_n^d$. In previous work, mixing times of this process for $p<p_c(mathbb{Z}^d)$ were obtained in the annealed setting where one averages over the dynamical percolation environment. Here we study exit times in the quenched setting, where we condition on a typical dynamical percolation environment. We obtain an upper bound for all $p$ which for $p<p_c$ matches the known lower bound.
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