On $(\mathbb Z/N\mathbb Z)^2$- Occupation Times, the Gaussian Free Field, and Random Interlacements
by Alain-Sol Sznitman  

Vol. 7 No. 4 (2012) P.565~P.602

ABSTRACT

We study the occupation times left by random walk on $(\mathbb Z/N\mathbb Z)^2$ at times either proportional to $N^2 \log N$ or much larger than $N^2 \log N$, and relate these random fields to the Gaussian free field pinned at the origin. Our results answer a question raised in [18] and mirror limit statements in [18] for the occupation times of large rods of size $N$ in $\mathbb Z^3$ by random interlacements at a level $u_N$ such that $u_N N^3$ is either proportional to $N^2 \log N$ or much larger than $N^2 \log N$.

KEYWORDS
Occupation times, Gaussian free field, random interlacements

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 60J27, 60F05, 60G15

MILESTONES

Received: 2011-07-15
Revised : 2011-12-23
Accepted: 2012-12-26

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