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

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 Comets et al. (Commun Math Phys 343:129–164, 2016). 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 Sznitman (Ann Math 171(3):2039–2087, 2010) 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.