arXiv subjects

Alexis Prévost

Publications and source records attributed to Alexis Prévost.

14 recordsLinked to original sources

arXiv2025

Arm exponent for the Gaussian free field on metric graphs in intermediate dimensions

We investigate the bond percolation model on transient weighted graphs ${G}$ induced by the excursion sets of the Gaussian free field on the corresponding metric graph. We assume that balls in ${G}$ have polynomial volume growth with growth exponent $α$ and that the Green's function for the random on ${G}$ exhibits a power law decay with exponent $ν$, in the regime $1\leq ν\leq \fracα{2}$. In particular, this includes the cases of ${G}=\mathbb{Z}^{3}$ for which $ν=1$, and ${G}= \mathbb{Z}^{4}$ for which $ν=\fracα{2}=2$. For all such graphs, we determine the leading-order asymptotic behavior for the critical one-arm probability, which we prove decays with distance $R$, like $R^{-\fracν{2}+o(1)}$. Our results are, in fact, more precise and yield logarithmic corrections when $ν>1$ as well as corrections of order $\log \log R$ when $ν=1$. We further obtain very sharp upper bounds on truncated two-point functions close to criticality, which are new when $ν>1$ and essentially optimal when $ν=1$. This extends previous results from arXiv:2101.05801 and arXiv:1807.11117.

math.PR↗

arXiv2024

Cluster volumes for the Gaussian free field on metric graphs

We study the volume of the critical clusters for the percolation of the level sets of the Gaussian free field on metric graphs. On $\mathbb{Z}^d$ below the upper-critical dimension $d=6$, we show that the largest such cluster in a box of side length $r$ has volume of order $r^{\frac{d+2}{2}}$, as conjectured by Werner in arXiv:2002.11487. This is in contrast to the mean-field regime $d>6$, where this volume is of order $r^4$. We further obtain precise asymptotic tails for the volume of the critical cluster of the origin, and a lower bound on the tail of the volume of the near-critical cluster of the origin below the upper-critical dimension. Our proof extends to any graph with polynomial volume growth and polynomial decay of the Green's function as long as the critical one-arm probability decays as the square root of the Green's function, which is satisfied in low enough dimension.

math.PR↗

arXiv2024

First passage percolation, local uniqueness for interlacements and capacity of random walk

The study of first passage percolation (FPP) for the random interlacements model has been initiated in arXiv:2112.12096, where it is shown that on $\mathbb{Z}^d$, $d\geq 3$, the FPP distance is comparable to the graph distance with high probability. In this article, we give an asymptotically sharp lower bound on this last probability, which additionally holds on a large class of transient graphs with polynomial volume growth and polynomial decay of the Green function. When considering the interlacement set in the low-intensity regime, the previous bound is in fact valid throughout the near-critical phase. In low dimension, we also present two applications of this FPP result: sharp large deviation bounds on local uniqueness of random interlacements, and on the capacity of a random walk in a ball.

math.PR↗

arXiv2024

Percolation for two-dimensional excursion clouds and the discrete Gaussian free field

We study percolative properties of excursion processes and the discrete Gaussian free field (dGFF) in the planar unit disk. We consider discrete excursion clouds, defined using random walks as a two-dimensional version of random interlacements, as well as its scaling limit, defined using Brownian motion. We prove that the critical parameters associated to vacant set percolation for the two models are the same and equal to $π/3.$ The value is obtained from a Schramm-Loewner evolution (SLE) computation. Via an isomorphism theorem, we use a generalization of the discrete result that also involves a loop soup (and an SLE computation) to show that the critical parameter associated to level set percolation for the dGFF is strictly positive and smaller than $\sqrt{π/2}.$ In particular this entails a strict inequality of the type $h_*<\sqrt{2u_*}$ between the critical percolation parameters of the dGFF and the two-dimensional excursion cloud. Similar strict inequalities are conjectured to hold in a general transient setup.

math.PR↗

arXiv2024

Critical one-arm probability for the metric Gaussian free field in low dimensions

We investigate the bond percolation model on transient weighted graphs ${G}$ induced by the excursion sets of the Gaussian free field on the corresponding metric graph. Under the sole assumption that its sign clusters do not percolate, we derive an extension of Lupu's formula for the two-point function at criticality. We then focus on the low-dimensional case $0< ν< \fracα{2}$, where $α$ governs the polynomial volume growth of $G$ and $ν$ the decay rate of the Green's function on $G$. In particular, this includes the benchmark case ${G}=\mathbb{Z}^3$, for which $α=3$ and $ν= α-2=1$. We prove under these assumptions that the critical one-arm probability decays with distance $R$ like $R^{-\fracν{2}}$, up to multiplicative constants.

math.PR↗

arXiv2024

Geometry of Gaussian free field sign clusters and random interlacements

For a large class of amenable transient weighted graphs $G$, we prove that the sign clusters of the Gaussian free field on $G$ fall into a regime of strong supercriticality, in which two infinite sign clusters dominate (one for each sign), and finite sign clusters are necessarily tiny, with overwhelming probability. Examples of graphs belonging to this class include regular lattices like $\mathbb{Z}^d$, for $d \geqslant 3$, but also more intricate geometries, such as Cayley graphs of suitably growing (finitely generated) non-Abelian groups, and cases in which random walks exhibit anomalous diffusive behavior, for instance various fractal graphs. As a consequence, we also show that the vacant set of random interlacements on these objects, introduced by Sznitman in arXiv:0704.2560, and which is intimately linked to the free field, contains an infinite connected component at small intensities. In particular, this result settles an open problem from arXiv:1010.1490.

math.PR↗

arXiv2024

Universality classes for percolation models with long-range correlations

We consider a class of percolation models where the local occupation variables have long-range correlations decaying as a power law $\sim r^{-a}$ at large distances $r$, for some $0< a< d$ where $d$ is the underlying spatial dimension. For several of these models, we present both, rigorous analytical results and matching simulations that determine the critical exponents characterizing the fixed point associated to their phase transition, which is of second order. The exact values we obtain are rational functions of the two parameters $a$ and $d$ alone, and do not depend on the specifics of the model.

cond-mat.stat-mech↗

arXiv2024

Generating Galton-Watson trees using random walks and percolation for the Gaussian free field

The study of Gaussian free field level sets on supercritical Galton-Watson trees has been initiated by Abächerli and Sznitman in Ann. Inst. Henri Poincaré Probab. Stat., 54(1):173--201, 2018. By means of entirely different tools, we continue this investigation and generalize their main result on the positivity of the associated percolation critical parameter $h_*$ to the setting of arbitrary supercritical offspring distribution and random conductances. A fortiori, this provides a positive answer to the open question raised at the end of the aforementioned article. What is more, in our setting it also establishes a rigorous proof of the physics literature mantra that positive correlations facilitate percolation when compared to the independent case. Our proof proceeds by constructing the Galton-Watson tree through an exploration via finite random walk trajectories. This exploration of the tree progressively unveils an infinite connected component in the random interlacements set on the tree, which is stable under small quenched noise. Using a Dynkin-type isomorphism theorem, we then infer the strict positivity of the critical parameter $ h_* .$ As a byproduct of our proof we obtain the transience of the random interlacement set and the level sets of the Gaussian free field above small positive levels on such Galton-Watson trees.

math.PR↗

arXiv2023

First passage percolation with long-range correlations and applications to random Schrödinger operators

We consider first passage percolation (FPP) with passage times generated by a general class of models with long-range correlations on $\mathbb{Z}^d$, $d\geq 2$, including discrete Gaussian free fields, Ginzburg-Landau $\nabla ϕ$ interface models or random interlacements as prominent examples. We show that the associated time constant is positive, the FPP distance is comparable to the Euclidean distance, and we obtain a shape theorem. We also present two applications for random conductance models (RCM) with possibly unbounded and strongly correlated conductances. Namely, we obtain a Gaussian heat kernel upper bound for RCMs with a general class of speed measures, and an exponential decay estimate for the Green function of RCMs with random killing measures.

math.PR↗

arXiv2023

Phase transition for the late points of random walk

Let $X$ be a random walk on the torus of side length $N$ in dimension $d\geq 3$ with uniform starting point, and $t_{\text{cov}}$ be the expected value of its cover time, which is the first time that $X$ has visited every vertex of the torus at least once. For $α> 0$, the set $\mathcal{L}^α$ of $α$-late points consists of those points not visited by $X$ at time $αt_{\text{cov}}$. We prove the existence of a value $α_* \in (\frac12,1)$ across which $\mathcal{L}^α$ trivialises as follows: for all $α> α_*$ and $ε\geq N^{-c}$ there exists a coupling of $\mathcal{L}^α$ and two occupation sets $\mathcal{B}^{α_\pm}$ of i.i.d. Bernoulli fields having the same density as $\mathcal{L}^{α\pm ε}$, which is asymptotic to $N^{-(α\pmε)d}$, with the property that the inclusion $ \mathcal{B}^{α_+} \subseteq \mathcal{L}^α \subseteq \mathcal{B}^{α_-}$ holds with high probability as $N \to \infty$. On the contrary, when $α\leq α_*$ there is no such coupling. Corresponding results also hold for the vacant set of random interlacements at high intensities. The transition at $α_*$ corresponds to the (dis-)appearance of `double-points' (i.e. neighboring pairs of points) in $\mathcal{L}^α$. We further describe the law of $\mathcal{L}^α$ for $α>\frac12$ by adding independent patterns to $\mathcal{B}^{α_{\pm}}$. In dimensions $d \geq 4$ these are exactly all two-point sets. When $d=3$ one must also include all connected three-point sets, but no other.

math.PR↗

arXiv2023

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

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.

math.PR↗

arXiv2023

Critical exponents for a percolation model on transient graphs

We consider the bond percolation problem on a transient weighted graph induced by the excursion sets of the Gaussian free field on the corresponding cable system. Owing to the continuity of this setup and the strong Markov property of the field on the one hand, and the links with potential theory for the associated diffusion on the other, we rigorously determine the behavior of various key quantities related to the (near-)critical regime for this model. In particular, our results apply in case the base graph is the three-dimensional cubic lattice. They unveil the values of the associated critical exponents, which are explicit but not mean-field and consistent with predictions from scaling theory below the upper-critical dimension.

math.PR↗

arXiv2022

Cluster capacity functionals and isomorphism theorems for Gaussian free fields

We investigate level sets of the Gaussian free field on continuous transient metric graphs $\tilde{\mathcal G}$ and study the capacity of its level set clusters. We prove, without any further assumption on the base graph $\mathcal{G}$, that the capacity of sign clusters on $\tilde{\mathcal G}$ is finite almost surely. This leads to a new and effective criterion to determine whether the sign clusters of the free field on $\tilde{\mathcal G}$ are bounded or not. It also elucidates why the critical parameter for percolation of level sets on $\tilde{\mathcal G}$ vanishes in most instances in the massless case and establishes the continuity of this phase transition in a wide range of cases, including all vertex-transitive graphs. When the sign clusters on $\tilde{\mathcal G}$ do not percolate, we further determine by means of isomorphism theory the exact law of the capacity of compact clusters at any height. Specifically, we derive this law from an extension of Sznitman's refinement of Lupu's recent isomorphism theorem relating the free field and random interlacements, proved along the way, and which holds under the sole assumption that sign clusters on $\tilde{\mathcal G}$ are bounded. Finally, we show that the law of the cluster capacity functionals obtained in this way actually characterizes the isomorphism theorem, i.e. the two are equivalent.

math.PR↗

arXiv2018

The sign clusters of the massless Gaussian free field percolate on $\mathbb{Z}^d$, $d \geqslant 3$ (and more)

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 settling an open question from arXiv:1202.5172. In particular, this implies that the sign clusters of the Gaussian free field percolate on $\mathbb{Z}^d$, for all $d\geqslant 3$. Among other things, our construction of an infinite cluster above small, but positive level $h$ involves random interlacements at level $u>0$, a random subset of $\mathbb{Z}^d$ with desirable percolative properties, introduced in arXiv:0704.2560 in a rather different context, a certain Dynkin-type isomorphism theorem relating random interlacements to the Gaussian free field, see arXiv:1111.4818, and a recent coupling from arXiv:1402.0298 of these two objects, lifted to a continuous metric graph structure over $\mathbb{Z}^d$.

math.PR↗