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Register a new userLevel-set percolation of the Gaussian free field on regular graphs I: Regular trees
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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].
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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].
In invasion percolation, the edges of successively maximal weight (the outlets) divide the invasion cluster into a chain of ponds separated by outlets. On the regular tree, the ponds are shown to grow exponentially, with law of large numbers, central limit theorem and large deviation results. The tail asymptotics for a fixed pond are also studied and are shown to be related to the asymptotics of a critical percolation cluster, with a logarithmic correction.
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 particular, the correlation length diverges exponentially, and the critical clusters are logarithmic fractals, whose area scales with the linear size as $A sim L^2 / sqrt{ln L}$. The two-point connectivity also decays as the log of the distance. We corroborate our theory by numerical simulations. Possible CFT interpretations are discussed.
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.
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.
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