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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.
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We introduce the model of two-dimensional continuous random interlacements, which is constructed using the Brownian trajectories conditioned on not hitting a fixed set (usually, a disk). This model yields the local picture of Wiener sausage on the torus around a late point. As such, it can be seen as a continuous analogue of discrete two-dimensional random interlacements [Comets, Popov, Vachkovskaia, 2016]. At the same time, one can view it as (restricted) Brownian loops through infinity. We establish a number of results analogous to these of [Comets, Popov, Vachkovskaia, 2016; Comets, Popov, 2016], as well as the results specific to the continuous case.
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].
In this paper we obtain a decoupling feature of the random interlacements process $mathcal{I}^u subset mathbb{Z}^d$, at level $u$, $dgeq 3$. More precisely, we show that the trace of the random interlacements process on two disjoint finite sets, $textsf{F}$ and its translated $textsf{F}+x$, can be coupled with high probability of success, when $|x|$ is large, with the trace of a process of independent excursions, which we call the noodle soup process. As a consequence, we obtain an upper bound on the covariance between two $[0,1]$-valued functions depending on the configuration of the random interlacements on $textsf{F}$ and $textsf{F}+x$, respectively. This improves a previous bound obtained by Sznitman in [12].
For a large class of amenable transient weighted graphs $G$, we prove that the sign clusters of the Gaussian free field on $G$ fall into a regime of strong supercriticality, in which two infinite sign clusters dominate (one for each sign), and finite sign clusters are necessarily tiny, with overwhelming probability. Examples of graphs belonging to this class include regular lattices like $mathbb{Z}^d$, for $d geqslant 3$, but also more intricate geometries, such as Cayley graphs of suitably growing (finitely generated) non-Abelian groups, and cases in which random walks exhibit anomalous diffusive behavior, for instance various fractal graphs. As a consequence, we also show that the vacant set of random interlacements on these objects, introduced by Sznitman in arXiv:0704.2560, and which is intimately linked to the free field, contains an infinite connected component at small intensities. In particular, this result settles an open problem from arXiv:1010.1490.
We consider the random interlacements process with intensity $u$ on ${mathbb Z}^d$, $dge 5$ (call it $I^u$), built from a Poisson point process on the space of doubly infinite nearest neighbor trajectories on ${mathbb Z}^d$. For $kge 3$ we want to determine the minimal number of trajectories from the point process that is needed to link together $k$ points in $mathcal I^u$. Let $$n(k,d):=lceil frac d 2 (k-1) rceil - (k-2).$$ We prove that almost surely given any $k$ points $x_1,...,x_kin mathcal I^u$, there is a sequence ofof $n(k,d)$ trajectories $gamma^1,...,gamma^{n(k,d)}$ from the underlying Poisson point process such that the union of their traces $bigcup_{i=1}^{n(k,d)}tr(gamma^{i})$ is a connected set containing $x_1,...,x_k$. Moreover we show that this result is sharp, i.e. that a.s. one can find $x_1,...,x_k in I^u$ that cannot be linked together by $n(k,d)-1$ trajectories.
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