Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userPercolation for the Gaussian free field on the cable system: counterexamples
70
0
0.0
(
0
)
Authors
Alexis Prevost
Ask ChatGPT about the research
No Arabic abstract
For massless vertex-transitive transient graphs, the percolation phase transition for the level sets of the Gaussian free field on the associated continuous cable system is particularly well understood, and in particular the associated critical parameter $widetilde{h}_*$ is always equal to zero. On general transient graphs, two weak conditions on the graph $mathcal{G}$ are given in arXiv:2101.05800, each of which implies one of the two inequalities $widetilde{h}_*leq0$ and $widetilde{h}_*geq0.$ In this article, we give two counterexamples to show that none of these two conditions are necessary, prove that the strict inequality $widetilde{h}_*<0$ is typical on massive graphs with bounded weights, and provide an example of a graph on which $widetilde{h}_*=infty.$ On the way, we obtain another characterization of random interlacements on massive graphs, as well as an isomorphism between the Gaussian free field and the Doob $mathit{mathbf{h}}$-transform of random interlacements, and between the two-dimensional pinned free field and random interlacements.
rate research
Read More
We consider the zero-average Gaussian free field on a certain class of finite $d$-regular graphs for fixed $dgeq 3$. This class includes $d$-regular expanders of large girth and typical realisations of random $d$-regular graphs. We show that the level set of the zero-average Gaussian free field above level $h$ exhibits a phase transition at level $h_star$, which agrees with the critical value for level-set percolation of the Gaussian free field on the infinite $d$-regular tree. More precisely, we show that, with probability tending to one as the size of the finite graphs tends to infinity, the level set above level $h$ does not contain any connected component of larger than logarithmic size whenever $h>h_star$, and on the contrary, whenever $h<h_star$, a linear fraction of the vertices is contained in connected components of the level set above level $h$ having a size of at least a small fractional power of the total size of the graph. It remains open whether in the supercritical phase $h<h_star$, as the size of the graphs tends to infinity, one observes the emergence of a (potentially unique) giant connected component of the level set above level $h$. The proofs in this article make use of results from the accompanying paper [AC1].
We study level-set percolation of the Gaussian free field on the infinite $d$-regular tree for fixed $dgeq 3$. Denoting by $h_star$ the critical value, we obtain the following results: for $h>h_star$ we derive estimates on conditional exponential moments of the size of a fixed connected component of the level set above level $h$; for $h<h_star$ we prove that the number of vertices connected over distance $k$ above level $h$ to a fixed vertex grows exponentially in $k$ with positive probability. Furthermore, we show that the percolation probability is a continuous function of the level $h$, at least away from the critical value $h_star$. Along the way we also obtain matching upper and lower bounds on the eigenfunctions involved in the spectral characterisation of the critical value $h_star$ and link the probability of a non-vanishing limit of the martingale used therein to the percolation probability. A number of the results derived here are applied in the accompanying paper [AC2].
The Rohde--Schramm theorem states that Schramm--Loewner Evolution with parameter $kappa$ (or SLE$_kappa$ for short) exists as a random curve, almost surely, if $kappa eq 8$. Here we give a new and concise proof of the result, based on the Liouville quantum gravity coupling (or reverse coupling) with a Gaussian free field. This transforms the problem of estimating the derivative of the Loewner flow into estimating certain correlated Gaussian free fields. While the correlation between these fields is not easy to understand, a surprisingly simple argument allows us to recover a derivative exponent first obtained by Rohde and Schramm, subsequently shown to be optimal by Lawler and Viklund, which then implies the Rohde--Schramm theorem.
In this paper we introduce the two-sided level-set for the two-dimensional discrete Gaussian free field. Then we investigate the chemical distance for the two-sided level-set percolation. Our result shows that the chemical distance should have dimension strictly larger than $1$, which in turn stimulates some tempting questions about the two-sided level-set.
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 on recent joint papers with O. Louidor and with J. Ding and S. Goswami. Still, new proofs of the tightness and distributional convergence of the centered DGFF maximum are presented that by-pass the use of the modified Branching Random Walk. The text contains a wealth of instructive exercises and a list of open questions and conjectures for future research.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
