Level-set percolation of the Gaussian free field on regular graphs II: finite expanders

Abächerli, Angelo; Černý, Jǐŕı

doi:10.1214/20-ejp532

Cited by 9 publications

(35 citation statements)

References 18 publications

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“…The results of [2] and [7] hold in fact for any deterministic sequence of large-girth expanders (conditions (I) and (II) in Proposition 1), which is w.h.p. the case for G n .…”

Section: Deterministic Regular Graphsmentioning

confidence: 84%

“…By Proposition 1.1 of [2], (I) and (II) imply that for some K 5 , K 6 > 0 and for n large enough, for all x, y ∈ V n ,…”

Section: Structure and Green Functionmentioning

confidence: 94%

“…In Section 5, we explore the connected component of a vertex in E ≥h ψ Gn . In Section 6, we prove (2). In Section 7, we prove (3).…”

Section: Plan Of the Rest Of The Papermentioning

confidence: 97%

“…for G n , so that for any sequence of dregular graphs (G n ) n≥1 satisfying them, Theorem 1 holds? In our proofs, averaging on the randomness of G n is a crucial ingredient to control the presence of cycles on large subgraphs of G n , and allows us to extend some arguments of [2], where ψ Gn is locally approximated by ϕ T d .…”

Section: Deterministic Regular Graphsmentioning

confidence: 99%

“…Abächerli and Cerny recently investigated the GFF on the d-regular tree T d [1], and the zeroaverage GFF on some d-regular graphs (large girth expanders) in a companion paper [2]. In this setting, many essential questions (such as the value of h , or the sharpness of the phase transition at h for the zero-average GFF) remain open.…”

Section: Introduction 1overviewmentioning

confidence: 99%

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Anatomy of a gaussian giant: supercritical level-sets of the free field on random regular graphs

Conchon--Kerjan¹

2021

Preprint

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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 ∈ (−∞, h ), where h is the level-set percolation threshold of the GFF on the d-regular tree T d . We prove that w.h.p as the number n of vertices diverges, the GFF has a unique giant connected component, where η(h) is the probability that the root percolates in the corresponding GFF level-set on T d . This gives a positive answer to the conjecture of [2] for most regular graphs. We also prove that the second largest component has size Θ(log n). Moreover, we show that C(n) 1 shares the following similarities with the giant component of the supercritical Erdős-Rényi random graph. First, the diameter and the typical distance between vertices are Θ(log n). Second, the 2-core and the kernel encompass a given positive proportion of the vertices. Third, the local structure is a branching process conditioned to survive, namely the level-set percolation cluster of the root in T d (in the Erdős-Rényi case, it is known to be a Galton-Watson tree with a Poisson distribution for the offspring).

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“…The results of [2] and [7] hold in fact for any deterministic sequence of large-girth expanders (conditions (I) and (II) in Proposition 1), which is w.h.p. the case for G n .…”

Section: Deterministic Regular Graphsmentioning

confidence: 84%

“…By Proposition 1.1 of [2], (I) and (II) imply that for some K 5 , K 6 > 0 and for n large enough, for all x, y ∈ V n ,…”

Section: Structure and Green Functionmentioning

confidence: 94%

“…In Section 5, we explore the connected component of a vertex in E ≥h ψ Gn . In Section 6, we prove (2). In Section 7, we prove (3).…”

Section: Plan Of the Rest Of The Papermentioning

confidence: 97%

Section: Deterministic Regular Graphsmentioning

confidence: 99%

Section: Introduction 1overviewmentioning

confidence: 99%

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Anatomy of a gaussian giant: supercritical level-sets of the free field on random regular graphs

Conchon--Kerjan¹

2021

Preprint

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Giant component for the supercritical level‐set percolation of the Gaussian free field on regular expander graphs

Černý

2023

Comm Pure Appl Math

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We consider the zero‐average Gaussian free field on a certain class of finite d‐regular graphs for fixed . 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 has a giant component in the whole supercritical phase, that is for all , with probability tending to one as the size of the graphs tends to infinity. In addition, we show that this component is unique. This significantly improves the result of [4], where it was shown that a linear fraction of vertices is in mesoscopic components if , and together with the description of the subcritical phase from [4] establishes a fully‐fledged percolation phase transition for the model.

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Gaussian level-set percolation on complex networks

Kühn

2024

Phys. Rev. E

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We present a solution of the problem of level-set percolation for multivariate Gaussians defined in terms of weighted graph Laplacians on complex networks. It is achieved using a cavity or message passing approach, which allows one to obtain the essential ingredient required for the solution, viz. a self-consistent determination of locally varying percolation probabilities. The cavity solution can be evaluated both for single large instances of locally treelike graphs, and in the thermodynamic limit of random graphs of finite mean degree in the configuration model class. The critical level hc of the percolation transition is obtained through the condition that the largest eigenvalue of a weighted version B of a nonbacktracking matrix satisfies λmax(B)|hc=1. We present level-dependent distributions of local percolation probabilities for Erdős-Rényi networks and and for networks with degree distributions described by power laws. We find that there is a strong correlation between marginal single-node variances of a massless multivariate Gaussian and local percolation probabilities at a given level h, which is nearly perfect at negative values h, but weakens as h↗0 for the system with power-law degree distribution, and generally also for negative values of h, if the multivariate Gaussian acquires a nonzero mass. The theoretical analysis simplifies in the case of random regular graphs with uniform edge weights of the weighted graph Laplacian of the system and uniform mass parameter of the Gaussian field. An asymptotic analysis reveals that for edge weights K=K(c)≡1 the critical percolation threshold hc decreases to 0, as the degree c of the random regular graph diverges. For K=(c)=1/c, however, the critical percolation threshold hc is shown to diverge as c→∞. Published by the American Physical Society 2024

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Level-set percolation of the Gaussian free field on regular graphs II: finite expanders

Cited by 9 publications

References 18 publications

Anatomy of a gaussian giant: supercritical level-sets of the free field on random regular graphs

Anatomy of a gaussian giant: supercritical level-sets of the free field on random regular graphs

Giant component for the supercritical level‐set percolation of the Gaussian free field on regular expander graphs

Gaussian level-set percolation on complex networks

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