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We consider dynamical percolation on the $d$-dimensional discrete torus of side length $n$, $mathbb{Z}_n^d$, where each edge refreshes its status at rate $mu=mu_nle 1/2$ to be open with probability $p$. We study random walk on the torus, where the walker moves at rate $1/(2d)$ along each open edge. In earlier work of two of the authors with A. Stauffer, it was shown that in the subcritical case $p<p_c(mathbb{Z}^d)$, the (annealed) mixing time of the walk is $Theta(n^2/mu)$, and it was conjectured that in the supercritical case $p>p_c(mathbb{Z}^d)$, the mixing time is $Theta(n^2+1/mu)$; here the implied constants depend only on $d$ and $p$. We prove a quenched (and hence annealed) version of this conjecture up to a poly-logarithmic factor under the assumption $theta(p)>1/2$. Our proof is based on percolation results (e.g., the Grimmett-Marstrand Theorem) and an analysis of the volume-biased evolving set process; the key point is that typically, the evolving set has a substantial intersection with the giant percolation cluster at many times. This allows us to use precise isoperimetric properties of the cluster (due to G. Pete) to infer rapid growth of the evolving set, which in turn yields the upper bound on the mixing time.
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 en
We study a random walk on $mathbb{F}_p$ defined by $X_{n+1}=1/X_n+varepsilon_{n+1}$ if $X_n eq 0$, and $X_{n+1}=varepsilon_{n+1}$ if $X_n=0$, where $varepsilon_{n+1}$ are independent and identically distributed. This can be seen as a non-linear analo
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 th
We study the evolution of a random walker on a conservative dynamic random environment composed of independent particles performing simple symmetric random walks, generalizing results of [16] to higher dimensions and more general transition kernels w
We consider a random walker in a dynamic random environment given by a system of independent simple symmetric random walks. We obtain ballisticity results under two types of perturbations: low particle density, and strong local drift on particles. Su