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Register a new userMoments and distribution of the local times of a transient random walk on $Z^d$
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Consider an arbitrary transient random walk on $Z^d$ with $dinN$. Pick $alphain[0,infty)$ and let $L_n(alpha)$ be the spatial sum of the $alpha$-th power of the $n$-step local times of the walk. Hence, $L_n(0)$ is the range, $L_n(1)=n+1$, and for integers $alpha$, $L_n(alpha)$ is the number of the $alpha$-fold self-intersections of the walk. We prove a strong law of large numbers for $L_n(alpha)$ as $ntoinfty$. Furthermore, we identify the asymptotic law of the local time in a random site uniformly distributed over the range. These results complement and contrast analogous results for recurrent walks in two dimensions recently derived by v{C}erny cite{Ce07}. Although these assertions are certainly known to experts, we could find no proof in the literature in this generality.
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We revisit an unpublished paper of Vervoort (2002) on the once reinforced random walk, and prove that this process is recurrent on any graph of the form $mathbb{Z}times Gamma$, with $Gamma$ a finite graph, for sufficiently large reinforcement parameter. We also obtain a shape theorem for the set of visited sites, and show that the fluctuations around this shape are of polynomial order. The proof involves sharp general estimates on the time spent on subgraphs of the ambiant graph which might be of independent interest.
We find formulas for the macroscopic Minkowski and Hausdorff dimensions of the range of an arbitrary transient walk in Z^d. This endeavor solves a problem of Barlow and Taylor (1991).
Fix $p>1$, not necessarily integer, with $p(d-2)<d$. We study the $p$-fold self-intersection local time of a simple random walk on the lattice $Z^d$ up to time $t$. This is the $p$-norm of the vector of the walkers local times, $ell_t$. We derive precise logarithmic asymptotics of the expectation of $exp{theta_t |ell_t|_p}$ for scales $theta_t>0$ that are bounded from above, possibly tending to zero. The speed is identified in terms of mixed powers of $t$ and $theta_t$, and the precise rate is characterized in terms of a variational formula, which is in close connection to the {it Gagliardo-Nirenberg inequality}. As a corollary, we obtain a large-deviation principle for $|ell_t|_p/(t r_t)$ for deviation functions $r_t$ satisfying $t r_tggE[|ell_t|_p]$. Informally, it turns out that the random walk homogeneously squeezes in a $t$-dependent box with diameter of order $ll t^{1/d}$ to produce the required amount of self-intersections. Our main tool is an upper bound for the joint density of the local times of the walk.
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.
We obtain sharp upper and lower bounds for the moderate deviations of the volume of the range of a random walk in dimension five and larger. Our results encompass two regimes: a Gaussian regime for small deviations, and a stretched exponential regime for larger deviations. In the latter regime, we show that conditioned on the moderate deviations event, the walk folds a small part of its range in a ball-like subset. Also, we provide new path properties, in dimension three as well. Besides the key role Newtonian capacity plays in this study, we introduce two original ideas, of general interest, which strengthen the approach developed in cite{AS}.
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