On pinned fields, interlacements, and random walk on $(\mathbb{Z}/N \mathbb{Z})^2$

TL;DR

This paper constructs a two-dimensional analogue of random interlacements using Poissonian soups of trajectories, linking it to pinned free fields and describing the local behavior of random walks on large tori avoiding a fixed point.

Contribution

It introduces a new model of pinned interlacements in two dimensions, connecting random walk traces, Poissonian trajectory ensembles, and the pinned free field.

Findings

Constructed a two-dimensional interlacement model via finite-volume approximations.

Linked the model to the pinned free field through a Ray-Knight type isomorphism.

Demonstrated the nontrivial limit of occupation fields as the torus size grows.

Abstract

We define two families of Poissonian soups of bidirectional trajectories on $\mathbb{Z}^2$, which can be seen to adequately describe the local picture of the trace left by a random walk on the two-dimensional torus $(\mathbb{Z}/N \mathbb{Z})^2$, started from the uniform distribution, run up to a time of order $(N\log N)^2$ and forced to avoid a fixed point. The local limit of the latter was recently established in arXiv:1502.03470. Our construction proceeds by considering, somewhat in the spirit of statistical mechanics, a sequence of finite-volume approximations, consisting of random walks avoiding the origin and killed at spatial scale $N$, either using Dirichlet boundary conditions, or by means of a suitably adjusted mass. By tuning the intensity $u$ of such walks with $N$, the occupation field can be seen to have a nontrivial limit, corresponding to that of the actual random walk. Our construction thus yields a two-dimensional analogue of the random interlacements model introduced in arXiv:0704.2560 in the transient case. It also links it to the pinned free field in $\mathbb{Z}^2$, by means of a (pinned) Ray-Knight type isomorphism theorem.