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Register a new userRemarks on the range and multiple range of random walk up to the time of exit
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We consider the scaling behavior of the range and $p$-multiple range, that is the number of points visited and the number of points visited exactly $pgeq 1$ times, of simple random walk on ${mathbb Z}^d$, for dimensions $dgeq 2$, up to time of exit from a domain $D_N$ of the form $D_N = ND$ where $Dsubset {mathbb R}^d$, as $Nuparrowinfty$. Recent papers have discussed connections of the range and related statistics with the Gaussian free field, identifying in particular that the distributional scaling limit for the range, in the case $D$ is a cube in $dgeq 3$, is proportional to the exit time of Brownian motion. The purpose of this note is to give a concise, different argument that the scaled range and multiple range, in a general setting in $dgeq 2$, both weakly converge to proportional exit times of Brownian motion from $D$, and that the corresponding limit moments are `polyharmonic, solving a hierarchy of Poisson equations.
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We study the scaling limit of the capacity of the range of a simple random walk on the integer lattice in dimension four. We establish a strong law of large numbers and a central limit theorem with a non-gaussian limit. The asymptotic behaviour is analogous to that found by Le Gall in 86 for the volume of the range in dimension two.
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).
We obtain estimates for large and moderate deviations for the capacity of the range of a random walk on $mathbb{Z}^d$, in dimension $dge 5$, both in the upward and downward directions. The results are analogous to those we obtained for the volume of the range in two companion papers [AS17, AS19]. Interestingly, the main steps of the strategy we developed for the latter apply in this seemingly different setting, yet the details of the analysis are different
We consider a discrete time simple symmetric random walk among Bernoulli obstacles on $mathbb{Z}^d$, $dgeq 2$, where the walk is killed when it hits an obstacle. It is known that conditioned on survival up to time $N$, the random walk range is asymptotically contained in a ball of radius $varrho_N=C N^{1/(d+2)}$ for any $dgeq 2$. For $d=2$, it is also known that the range asymptotically contains a ball of radius $(1-epsilon)varrho_N$ for any $epsilon>0$, while the case $dgeq 3$ remains open. We complete the picture by showing that for any $dgeq 2$, the random walk range asymptotically contains a ball of radius $varrho_N-varrho_N^epsilon$ for some $epsilon in (0,1)$. Furthermore, we show that its boundary is of size at most $varrho_N^{d-1}(log varrho_N)^a$ for some $a>0$.
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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