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We prove the existence of non-trivial phase transitions for the intersection of two independent random interlacements and the complement of the intersection. Some asymptotic results about the phase curves are also obtained. Moreover, we show that at least one of these two sets percolates in high dimensions.
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 de
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
We prove the existence of the intersection local time for two independent, d -dimensional fractional Brownian motions with the same Hurst parameter H. Assume d greater or equal to 2, then the intersection local time exists if and only if Hd<2.
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 to
These lecture notes offer a gentle introduction to the two-dimensional Discrete Gaussian Free Field with particular attention paid to the scaling limits of the level sets at heights proportional to the absolute maximum. The bulk of the text is based