The vacant set of two-dimensional critical random interlacement is infinite

For the model of two-dimensional random interlacements in the critical regime (i.e., $\alpha=1$), we prove that the vacant set is a.s.\ infinite, thus solving an open problem from arXiv:1502.03470. Also, we prove that the entrance measure of simple random walk on annular domains has certain regularity properties; this result is useful when dealing with soft local times for excursion processes.Comment: 38 pages, 3 figures; to appear in The Annals of Probabilit

Soft local times and decoupling of random interlacements

In this paper we establish a decoupling feature of the random interlacement process I^u in Z^d, at level u, for d \geq 3. Roughly speaking, we show that observations of I^u restricted to two disjoint subsets A_1 and A_2 of Z^d are approximately independent, once we add a sprinkling to the process I^u by slightly increasing the parameter u. Our results differ from previous ones in that we allow the mutual distance between the sets A_1 and A_2 to be much smaller than their diameters. We then provide an important application of this decoupling for which such flexibility is crucial. More precisely, we prove that, above a certain critical threshold u**, the probability of having long paths that avoid I^u is exponentially small, with logarithmic corrections for d=3. To obtain the above decoupling, we first develop a general method for comparing the trace left by two Markov chains on the same state space. This method is based in what we call the soft local time of a chain. In another crucial step towards our main result, we also prove that any discrete set can be "smoothened" into a slightly enlarged discrete set, for which its equilibrium measure behaves in a regular way. Both these auxiliary results are interesting in themselves and are presented independently from the rest of the paper.Comment: 10 figure

On multidimensional branching random walks in random environment

We study branching random walks in random i.i.d. environment in $\Z^d, d \geq 1$. For this model, the population size cannot decrease, and a natural definition of recurrence is introduced. We prove a dichotomy for recurrence/transience, depending only on the support of the environmental law. We give sufficient conditions for recurrence and for transience. In the recurrent case, we study the asymptotics of the tail of the distribution of the hitting times and prove a shape theorem for the set of lattice sites which are visited up to a large time