Abstract
Abstract
We consider a simple random walk on
$\mathbb{Z}^d$
started at the origin and stopped on its first exit time from
$({-}L,L)^d \cap \mathbb{Z}^d$
. Write L in the form
$L = m N$
with
$m = m(N)$
and N an integer going to infinity in such a way that
$L^2 \sim A N^d$
for some real constant
$A \gt 0$
. Our main result is that for
$d \ge 3$
, the projection of the stopped trajectory to the N-torus locally converges, away from the origin, to an interlacement process at level
$A d \sigma_1$
, where
$\sigma_1$
is the exit time of a Brownian motion from the unit cube
$({-}1,1)^d$
that is independent of the interlacement process. The above problem is a variation on results of Windisch (2008) and Sznitman (2009).
Publisher
Cambridge University Press (CUP)
Subject
Applied Mathematics,Statistics and Probability