Critical exponents for a percolation model on transient graphs

Drewitz, Alexander; Prévost, Alexis; Rodriguez, Pierre-François

doi:10.48550/arxiv.2101.05801

Cited by 4 publications

(6 citation statements)

References 38 publications

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“…Let us mention that both decays (1.16) and (1.17) have been proved for the discrete GFF on certain graphs [28,14,11,34,27,26]. We now discuss some aspects of our global comparison theorem.…”

Section: Open Questionsmentioning

confidence: 83%

Sharp phase transition for Gaussian percolation in all dimensions

Severo¹

2021

Preprint

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We consider the level-sets of continuous Gaussian fields on R d above a certain level −ℓ ∈ R, which defines a percolation model as ℓ varies. We assume that the covariance kernel satisfies certain regularity, symmetry and positivity conditions as well as a polynomial decay with exponent greater than d (in particular, this includes the Bargmann-Fock field). Under these assumptions, we prove that the model undergoes a sharp phase transition around its critical point ℓ c . More precisely, we show that connection probabilities decay exponentially for ℓ < ℓ c and percolation occurs in sufficiently thick 2D slabs for ℓ > ℓ c . This extends results recently obtained in dimension d = 2 to arbitrary dimensions through completely different techniques. The result follows from a global comparison with a truncated (i.e. with finite range of dependence) and discretized (i.e. defined on the lattice εZ d ) version of the model, which may be of independent interest. The proof of this comparison relies on an interpolation scheme that integrates out the long-range and infinitesimal correlations of the model while compensating them with a slight change in the parameter ℓ.

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Section: Open Questionsmentioning

confidence: 83%

Sharp phase transition for Gaussian percolation in all dimensions

Severo¹

2021

Preprint

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“…We will then use killed or surviving interlacements to give a first illustration of the independent interest of these results, see Corollaries 3.4, 6.9 and 6.11. We refer to Section 6 in [13] for an example of a proof where killed interlacements and killed-surviving interlacements play an essential role.…”

Section: Notation and Definitionmentioning

confidence: 99%

“…This model often undergoes a phase transition at level zero, which was first proved in [22] on Z d , d ≥ 3, then on trees in [35] and [1], and later on a large class of transient graphs, possibly with positive killing measure, in [12]. In many cases, this phase transition is also particularly well understood in the near-critical regime, see [9] on Z d , d ≥ 3, or the recent paper [13] for additional results on general graphs. We will prove that this behaviour of the phase transition, although typical, cannot be extended to any transient weighted graph.…”

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confidence: 92%

Percolation for the Gaussian free field on the cable system: counterexamples

Prévost¹

2021

Preprint

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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 h * is always equal to zero. On general transient graphs, two weak conditions on the graph G are given in [12], each of which implies one of the two inequalities h * ≤ 0 and h * ≥ 0. In this article, we give two counterexamples to show that none of these two conditions are necessary, prove that the strict inequality h * < 0 is typical on massive graphs with bounded weights, and provide an example of a graph on which h * = ∞. 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 h-transform of random interlacements, and between the two-dimensional pinned free field and random interlacements. Contents

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“…Interestingly, in the case of the cable graph on Z d the corresponding function θG 0 is explicit. The critical level is 0 and θG 0 (h) = 2Φ(h ∧ 0), for h ∈ R, where Φ denotes the distribution function of a centered Gaussian variable with variance g(0, 0), see Corollary 2.1 of [12]. However, in the case of Z d , d = 3, the simulations in Figure 4 of [21] suggest a behavior of θ G 0 close to the critical level h * different from that of θG 0 near the critical level 0.…”

Section: A Appendix: Resonance Sets and I-familiesmentioning

confidence: 99%

On the cost of the bubble set for random interlacements

Sznitman

2021

Preprint

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The main focus of this article concerns the strongly percolative regime of the vacant set of random interlacements on Z d , d ≥ 3. We investigate the occurrence in a large box of an excessive fraction of sites that get disconnected by the interlacements from the boundary of a concentric box of double size. The results significantly improve our findings in [29]. In particular, if, as expected, the critical levels for percolation and for strong percolation of the vacant set of random interlacements coincide, the asymptotic upper bound that we derive here matches in principal order a previously known lower bound. A challenging difficulty revolves around the possible occurrence of droplets that could get secluded by the random interlacements and thus contribute to the excess of disconnected sites, somewhat in the spirit of the Wulff droplets for Bernoulli percolation or for the Ising model. This feature is reflected in the present context by the so-called bubble set, a possibly quite irregular random set. A pivotal progress in this work has to do with the improved construction of a coarse grained random set accounting for the cost of the bubble set. This construction heavily draws both on the method of enlargement of obstacles originally developed in the mid-nineties in the context of Brownian motion in a Poissonian potential in [26], [27], and on the resonance sets recently introduced by Nitzschner and the author in [23] and further developed in a discrete set-up by Chiarini and Nitzschner in [9].

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Critical exponents for a percolation model on transient graphs

Cited by 4 publications

References 38 publications

Sharp phase transition for Gaussian percolation in all dimensions

Sharp phase transition for Gaussian percolation in all dimensions

Percolation for the Gaussian free field on the cable system: counterexamples

On the cost of the bubble set for random interlacements

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