Related papers: High-dimensional asymptotics for percolation of Ga…
For the Gaussian free field on a $(d + 1)$-regular tree with $d \geq 2$, we study the percolative properties of its level sets in the critical and the near-critical regime. In particular, we show the continuity of the percolation…
We investigate the percolation phase transition for level sets of the Gaussian free field on $\mathbb{Z}^d$, with $d\geqslant 3$, and prove that the corresponding critical parameter $h_*(d)$ is strictly positive for all $d\geqslant3$, thus…
We consider level-set percolation for the Gaussian free field on Z^d, with d bigger or equal to 3, and prove that there is a non-trivial critical level h_* such that for h > h_*, the excursion set above level h does not percolate, and for h…
The vacant set of random interlacements on ${\mathbb{Z}}^d$, $d\ge3$, has nontrivial percolative properties. It is known from Sznitman [Ann. Math. 171 (2010) 2039--2087], Sidoravicius and Sznitman [Comm. Pure Appl. Math. 62 (2009) 831--858]…
We consider upper level-sets of the Gaussian free field on $\mathbb Z^d$, for $d\geq 3$, above a given real-valued height parameter $h$. As $h$ varies, this defines a canonical percolation model with strong, algebraically decaying…
We study level-set percolation of the Gaussian free field on the infinite $d$-regular tree for fixed $d\geq 3$. Denoting by $h_\star$ the critical value, we obtain the following results: for $h>h_\star$ we derive estimates on conditional…
We consider a general enough set-up and obtain a refinement of the coupling between the Gaussian free field and random interlacements recently constructed by Titus Lupu in arXiv:1402.0298. We apply our results to level-set percolation of…
We investigate level-set percolation of the Gaussian free field on transient trees, for instance on super-critical Galton-Watson trees conditioned on non-extinction. Recently developed Dynkin-type isomorphism theorems provide a comparison…
We study the level-set percolation of the Gaussian free field on Z^d, d bigger or equal to 3. We consider a level alpha such that the excursion-set of the Gaussian free field above alpha percolates. We derive large deviation estimates on…
We consider the zero-average Gaussian free field on a certain class of finite $d$-regular graphs for fixed $d\geq 3$. This class includes $d$-regular expanders of large girth and typical realisations of random $d$-regular graphs. We show…
We investigate the percolative properties of the vacant set left by random interlacements on Z^d, when d is large. A non-negative parameter u controls the density of random interlacements on Z^d. It is known from arXiv:0704.2560, and…
We consider $Z^d$, with d bigger or equal to three. We investigate the vacant set of random interlacements in the strongly percolative regime, the vacant set of the simple random walk, and the excursion set above a given level of the…
We consider continuous-time random interlacements on Z^d, d greater or equal to 3, and investigate the percolation model where a site x of Z^d is occupied if the total amount of time spent at x by all the trajectories of the interlacement…
For $d \geq 3$ we obtain an approximation of the zero-average Gaussian free field on the discrete $d$-dimensional torus of large side length $N$ by the Gaussian free field on $\mathbb Z^d$, valid in boxes of roughly side length $N -…
The nature of level set percolation in the two-dimension Gaussian Free Field has been an elusive question. Using a loop-model mapping, we show that there is a nontrivial percolation transition, and characterize the critical point. In…
We consider the zero-average Gaussian free field on a certain class of finite $d$-regular graphs for fixed $d\ge 3$. This class includes $d$-regular expanders of large girth and typical realisations of random $d$-regular graphs. We show…
We prove that cluster observables of level-sets of the Gaussian free field on the hypercubic lattice $\mathbb{Z}^d$, $d\geq3$, are analytic on the whole off-critical regime $\mathbb{R}\setminus\{h_*\}$. This result concerns in particular…
We study an asymptotic expansion of the critical point for the nearest-neighbor oriented percolation on $\mathbb Z^d$ in powers of $d^{-1}$ as $d\rightarrow \infty$. The proof relies heavily on the lace expansion.
In this paper, we study the level-set of the zero-average Gaussian Free Field on a uniform random $d$-regular graph above an arbitrary level $h\in (-\infty, h_{\star})$, where $h_{\star}$ is the level-set percolation threshold of the GFF on…
We consider a Poisson point process on the space of lines in R^d, where a multiplicative factor u>0 of the intensity measure determines the density of lines. Each line in the process is taken as the axis of a bi-infinite cylinder of radius…