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

• Published in: Probability theory and related fields 2019-04, Vol.173 (3-4), p.1265-1299
• Main Author: Rodriguez, Pierre-François
• Format: Article
• Language: English
• Subjects: Comets; Dirichlet problem; Isomorphism; Mathematics; Probability; Random walk; Random walk theory; Soups; Statistical mechanics; Toruses; Two dimensional models
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-018-0851-z

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.