For $dge 3$ we construct a new coupling of the trace left by a random walk on a large $d$-dimensional discrete torus with the random interlacements on $mathbb Z^d$. This coupling has the advantage of working up to macroscopic subsets of the torus. As
an application, we show a sharp phase transition for the diameter of the component of the vacant set on the torus containing a given point. The threshold where this phase transition takes place coincides with the critical value $u_*(d)$ of random interlacements on $mathbb Z^d$. Our main tool is a variant of the soft-local time coupling technique of [PT12].
We consider the model of random interlacements on transient graphs, which was first introduced by Sznitman [Ann. of Math. (2) (2010) 171 2039-2087] for the special case of ${mathbb{Z}}^d$ (with $dgeq3$). In Sznitman [Ann. of Math. (2) (2010) 171 2039
-2087], it was shown that on ${mathbb{Z}}^d$: for any intensity $u>0$, the interlacement set is almost surely connected. The main result of this paper says that for transient, transitive graphs, the above property holds if and only if the graph is amenable. In particular, we show that in nonamenable transitive graphs, for small values of the intensity u the interlacement set has infinitely many infinite clusters. We also provide examples of nonamenable transitive graphs, for which the interlacement set becomes connected for large values of u. Finally, we establish the monotonicity of the transition between the disconnected and the connected phases, providing the uniqueness of the critical value $u_c$ where this transition occurs.
We study the trajectory of a simple random walk on a d-regular graph with d>2 and locally tree-like structure as the number n of vertices grows. Examples of such graphs include random d-regular graphs and large girth expanders. For these graphs, we i
nvestigate percolative properties of the set of vertices not visited by the walk until time un, where u>0 is a fixed positive parameter. We show that this so-called vacant set exhibits a phase transition in u in the following sense: there exists an explicitly computable threshold u* such that, with high probability as n grows, if u<u*, then the largest component of the vacant set has a volume of order n, and if u>u*, then it has a volume of order log(n). The critical value u* coincides with the critical intensity of a random interlacement process (introduced by Sznitman [arXiv:0704.2560]) on a d-regular tree. We also show that the random interlacement model describes the structure of the vacant set in local neighbourhoods.
