  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.
