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We define two families of Poissonian soups of bidirectional trajectories on$\mathbb{Z}^2$, which can be seen to adequately describe the local picture ofthe 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 oforder $(N\log N)^2$ and forced to avoid a fixed point. The local limit of thelatter was recently established in arXiv:1502.03470. Our construction proceedsby considering, somewhat in the spirit of statistical mechanics, a sequence offinite-volume approximations, consisting of random walks avoiding the originand killed at spatial scale $N$, either using Dirichlet boundary conditions, orby means of a suitably adjusted mass. By tuning the intensity $u$ of such walkswith $N$, the occupation field can be seen to have a nontrivial limit,corresponding to that of the actual random walk. Our construction thus yields atwo-dimensional analogue of the random interlacements model introduced inarXiv:0704.2560 in the transient case. It also links it to the pinned freefield in $\mathbb{Z}^2$, by means of a (pinned) Ray-Knight type isomorphismtheorem.

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