arXiv subjects

Pierre Nolin

Publications and source records attributed to Pierre Nolin.

At least 19 recordsLinked to original sources

arXiv2025

Arm events in critical planar loop soups

We establish up-to-constants estimates for arm events in the Brownian loop soup on the 2D metric graph associated with the square lattice. More specifically, we consider two natural geometric events: first, ``bulk'' four-arm events, corresponding to two large connected components of loops getting close to each other; and then, two-arm events in the half-plane, used to estimate the probability that a cluster of loops approaches the boundary. Our proof relies on an estimate by Lupu-Werner [Probab. Theory Related Fields 171(3):775-818, 2018], thanks to the well-known coupling between the loop soup and the Gaussian free field on the metric graph [Lecture Notes in Mathematics, volume 2026, 2011] and [Ann. Probab. 44(3):2117-2146, 2016]. As a consequence, we also obtain up-to-constant upper bounds for the corresponding arm events in the random walk loop soup on the square lattice. In this way, we verify Assumptions 5.7 and 5.11 in arXiv:2409.16273: in a box with side length $N$, this implies the existence of crossings where the Gaussian free field remains below $a\sqrt{\log \log N}$ in absolute value, for some constant $a > 0$ large enough.

math.PR↗

arXiv2025

Up-to-constants estimates on four-arm events for simple conformal loop ensemble

We prove up-to-constants estimates for a general class of four-arm events in simple conformal loop ensembles, i.e. CLE$_\kappa$ for $\kappa\in (8/3,4]$. The four-arm events that we consider can be created by either one or two loops, with no constraint on the topology of the crossings. Our result is a key input in our series of works arxiv:2409.16230 and arxiv:2409.16273 on percolation of the two-sided level sets in the discrete Gaussian free field (and level sets in the occupation field of the random walk loop soup). In order to get rid of all constraints on the topology of the crossings, we rely on the Brownian loop-soup representation of simple CLE [Ann. Math. 176 (2012) 1827-1917], and a "cluster version" of a separation lemma for the Brownian loop soup. As a corollary, we also obtain up-to-constants estimates for a general version of four-arm events for SLE$_\kappa$ for $\kappa\in (8/3,4]$. This fixes (in the case of four arms and $\kappa\in(8/3,4]$) an essential gap in [Ann. Probab. 46 (2018) 2863-2907] and improves some estimates therein.

math.PR↗

arXiv2024

Backbone exponent and annulus crossing probability for planar percolation

We report the recent derivation of the backbone exponent for 2D percolation. In contrast to previously known exactly solved percolation exponents, the backbone exponent is a transcendental number, which is a root of an elementary equation. We also report an exact formula for the probability that there are two disjoint paths of the same color crossing an annulus. The backbone exponent captures the leading asymptotic, while the other roots of the elementary equation capture the asymptotic of the remaining terms. This suggests that the backbone exponent is part of a conformal field theory (CFT) whose bulk spectrum contains this set of roots. Our approach is based on the coupling between SLE curves and Liouville quantum gravity (LQG), and the integrability of Liouville CFT that governs the LQG surfaces.

cond-mat.stat-mech↗

arXiv2024

Percolation of discrete GFF in dimension two I. Arm events in the random walk loop soup

In this work, which is the first part of a series of two papers, we study the random walk loop soup in dimension two. More specifically, we estimate the probability that two large connected components of loops come close to each other, in the subcritical and critical regimes. The associated four-arm event can be estimated in terms of exponents computed in the Brownian loop soup, relying on the connection between this continuous process and conformal loop ensembles (with parameter $\kappa \in (8/3,4]$). Along the way, we need to develop several useful tools for the loop soup, based on separation for random walks and surgery for loops, such as a "locality" property and quasi-multiplicativity. The results established here then play a key role in a second paper, in particular to study the connectivity properties of level sets in the random walk loop soup and in the discrete Gaussian free field.

math.PR↗

arXiv2024

Percolation of discrete GFF in dimension two II. Connectivity properties of two-sided level sets

We study percolation of two-sided level sets for the discrete Gaussian free field (DGFF) in 2D. For a DGFF $\varphi$ defined in a box $B_N$ with side length $N$, for $C$ large enough, there exist low crossings in the set of vertices $z$ where $|\varphi(z)|\le C \sqrt{\log \log N}$, with probability tending to $1$ as $N \to \infty$, while the average and the maximum of $\varphi$ are of order $\sqrt{\log N}$ and $\log N$, respectively. As a consequence, we also obtain connectivity properties of the set of thick points of a random walk. We rely on an isomorphism between the DGFF and the random walk loop soup (RWLS) with critical intensity $\alpha=1/2$, and further extend our study to the occupation field of the RWLS for all subcritical intensities $\alpha\in(0,1/2)$. For the RWLS in $B_N$, we show that for $\lambda$ large enough, there exist low crossings of $B_N$, remaining below $\lambda$, even though the average occupation time is of order $\log N$. Our results thus uncover a non-trivial phase-transition for this highly-dependent percolation model. For both the DGFF and the occupation field of the RWLS, we further show that such low crossings can be found in the "carpet" of the RWLS - the set of vertices which are not in the interior of any cluster of loops. This work is the second part of a series of two papers. It relies heavily on tools and techniques developed for the RWLS in the first part, especially surgery arguments on loops, which were made possible by a separation result in the RWLS. This allowed us, in that companion paper, to derive several useful properties such as quasi-multiplicativity, and obtain a precise upper bound for the probability that two large connected components of loops "almost touch", which is instrumental here.

math.PR↗

arXiv2024

Two-dimensional forest fires with boundary ignitions

In the classical Drossel-Schwabl forest fire process, vertices of a lattice become occupied at rate $1$, and they are hit by lightning at some tiny rate $\zeta > 0$, which causes entire connected components to burn. In this paper, we study a variant where fires are coming from the boundary of the forest instead. In particular we prove that, for the case without recoveries where the forest is an $N \times N$ box in the triangular lattice, the probability that the center of the box gets burnt tends to $0$ as $N \rightarrow \infty$ (but substantially slower than the one-arm probability of critical Bernoulli percolation). And, for the case where the forest is the upper-half plane, we show (still for the version without recoveries) that no infinite occupied cluster emerges. We also discuss analogs of some of these results for the corresponding models with recoveries, and explain how our results and proofs give valuable insight on a process considered earlier by Graf.

math.PR↗

arXiv2023

Backbone exponent for two-dimensional percolation

We derive an exact expression for the celebrated backbone exponent for Bernoulli percolation in dimension two at criticality. It turns out to be a root of an elementary function. Contrary to previously known arm exponents for this model, which are all rational, it has a transcendental value. Our derivation relies on the connection to the SLE$_\kappa$ bubble measure, the coupling between SLE and Liouville quantum gravity, and the integrability of Liouville conformal field theory. Along the way, we derive a formula not only for $\kappa=6$ (corresponding to percolation), but for all $\kappa \in (4,8)$.

math.PR↗

arXiv2022

A 2D forest fire process beyond the critical time

We study forest fire processes in two dimensions. On a given planar lattice, vertices independently switch from vacant to occupied at rate $1$ (initially they are all vacant), and any connected component "is burnt" (its vertices become instantaneously vacant) as soon as its cardinality crosses a (typically large) threshold $N$, the parameter of the model. Our analysis provides a detailed description, as $N \to \infty$, of the process near and beyond the critical time $t_c$ (at which an infinite cluster would arise in the absence of fires). In particular we prove a somewhat counterintuitive result: there exists $\delta > 0$ such that with high probability, the origin does not burn before time $t_c + \delta$. This provides a negative answer to Open Problem 4.1 of van den Berg and Brouwer [Comm. Math. Phys., 2006]. Informally speaking, the result can be explained in terms of the emergence of fire lanes, whose total density is negligible (as $N \to \infty$), but which nevertheless are sufficiently robust with respect to recoveries. We expect that such a behavior also holds for the classical Drossel-Schwabl model. A large part of this paper is devoted to analyzing recoveries during the interval $[t_c, t_c + \delta]$. These recoveries do have a "microscopic" effect, but it turns out that their combined influence on macroscopic scales (and in fact on relevant "mesoscopic" scales) vanishes as $N \to \infty$. In order to prove this, we use key ideas of Kiss, Manolescu and Sidoravicius [Ann. Probab., 2015], introducing a suitable induction argument to extend and strengthen their results. We then use it to prove that a deconcentration result in our earlier joint work with Kiss on volume-frozen percolation also holds for the forest fire process. As we explain, significant additional difficulties arise here, since recoveries destroy the nice spatial Markov property of frozen percolation.

math.PR↗

arXiv2021

Near-critical avalanches in 2D frozen percolation and forest fires

We study two closely related processes on the triangular lattice: frozen percolation, where connected components of occupied vertices freeze (they stop growing) as soon as they contain at least $N$ vertices, and forest fire processes, where connected components burn (they become entirely vacant) at rate $\zeta > 0$. In this paper, we prove that when the density of occupied sites approaches the critical threshold for Bernoulli percolation, both processes display a striking phenomenon: the appearance of near-critical "avalanches". More specifically, we analyze the avalanches, all the way up to the natural characteristic scale of each model, which constitutes an important step toward understanding the self-organized critical behavior of such processes. For frozen percolation, we show in particular that the number of frozen clusters surrounding a given vertex is asymptotically equivalent to $(\log(96/5))^{-1} \log \log N$ as $N \to \infty$. A similar mechanism underlies forest fires, enabling us to obtain an analogous result for these processes, but with substantially more work: the number of burnt clusters is equivalent to $(\log(96/41))^{-1} \log \log (\zeta^{-1})$ as $\zeta \searrow 0$. Moreover, almost all of these clusters have a volume $\zeta^{- 91/55 + o(1)}$. For forest fires, the percolation process with impurities introduced in arXiv:1810.08181 plays a crucial role in our proofs, and we extend the results in that paper, up to a positive density of impurities. In addition, we develop a novel exploration procedure to couple full-plane forest fires with processes in finite but large enough (compared to the characteristic scale) domains.

math.PR↗

arXiv2020

On the four-arm exponent for 2D percolation at criticality

For two-dimensional percolation at criticality, we discuss the inequality $\alpha_4 > 1$ for the polychromatic four-arm exponent (and stronger versions, the strongest so far being $\alpha_4 \geq 1 + \frac{\alpha_2}{2}$, where $\alpha_2$ denotes the two-arm exponent). We first briefly discuss five proofs (some of them implicit and not self-contained) from the literature. Then we observe that, by combining two of them, one gets a completely self-contained (and yet quite short) proof.

math.PR↗

arXiv2019

No exceptional words for Bernoulli percolation

Benjamini and Kesten introduced in 1995 the problem of embedding infinite binary sequences into a Bernoulli percolation configuration, known as "percolation of words". We give a positive answer to their Open Problem 2: almost surely, all words are seen for site percolation on Z^3 with parameter p = 1/2. We also extend this result in various directions, proving the same result for any dimension d at least three and for any value p in the interval (p_c(Z^d), 1 - p_c(Z^d)), and for restrictions to slabs. Finally, we provide an explicit estimate on the probability to find all words starting from a finite box.

math.PR↗

arXiv2018

Near-critical percolation with heavy-tailed impurities, forest fires and frozen percolation

Consider critical site percolation on a "nice" planar lattice: each vertex is occupied with probability $p = p_c$, and vacant with probability $1 - p_c$. Now, suppose that additional vacancies ("holes", or "impurities") are created, independently, with some small probability, i.e. the parameter $p_c$ is replaced by $p_c - \varepsilon$, for some small $\varepsilon > 0$. A celebrated result by Kesten says, informally speaking, that on scales below the characteristic length $L(p_c - \varepsilon)$, the connection probabilities remain of the same order as before. We prove a substantial and subtle generalization to the case where the impurities are not only microscopic, but allowed to be "mesoscopic". This generalization, which is also interesting in itself, was motivated by our study of models of forest fires (or epidemics). In these models, all vertices are initially vacant, and then become occupied at rate $1$. If an occupied vertex is hit by lightning, which occurs at a (typically very small) rate $\zeta$, its entire occupied cluster burns immediately, so that all its vertices become vacant. Our results for percolation with impurities turn out to be crucial for analyzing the behavior of these forest fire models near and beyond the critical time (i.e. the time after which, in a forest without fires, an infinite cluster of trees emerges). In particular, we prove (so far, for the case when burnt trees do not recover) the existence of a sequence of "exceptional scales" (functions of $\zeta$). For forests on boxes with such side lengths, the impact of fires does not vanish in the limit as $\zeta \searrow 0$.

math.PR↗

arXiv2016

Boundary rules and breaking of self-organized criticality in 2D frozen percolation

We study frozen percolation on the (planar) triangular lattice, where connected components stop growing ("freeze") as soon as their "size" becomes at least $N$, for some parameter $N \geq 1$. The size of a connected component can be measured in several natural ways, and we consider the two particular cases of diameter and volume (i.e. number of sites). Diameter-frozen and volume-frozen percolation have been studied in previous works, and they display radically different behaviors. These works adopt the rule that the boundary of a frozen cluster stays vacant forever, and we investigate the influence of these "boundary conditions" in the present paper. We prove the (somewhat surprising) result that they strongly matter in the diameter case, and we discuss briefly the volume case.

math.PR↗

arXiv2015

Two-dimensional volume-frozen percolation: deconcentration and prevalence of mesoscopic clusters

Frozen percolation on the binary tree was introduced by Aldous around fifteen years ago, inspired by sol-gel transitions. We investigate a version of the model on the triangular lattice, where connected components stop growing ("freeze") as soon as they contain at least $N$ vertices, for some parameter $N \geq 1$. This process has a substantially different behavior from the diameter-frozen process, studied in previous works: in particular, we show that many (more and more as $N \to \infty$) frozen clusters surrounding the origin appear successively, each new cluster having a diameter much smaller than the previous one. This separation of scales is instrumental, and it helps to approximate the process in sufficiently large (but not too large), as a function of $N$, finite domains by a Markov chain. This allows us to establish a deconcentration property for the sizes of the holes of the frozen clusters around the origin. For the full-plane process, we then show that it can be compared to the process in large finite domains, so that the deconcentration property also holds in this case. In particular, we obtain that with high probability (as $N \to \infty$), the origin does not belong to a frozen cluster in the final configuration. This work requires new properties for near-critical percolation, which we develop along the way, and which are interesting in their own right: in particular, an asymptotic formula involving the percolation probability $\theta(p)$ as $p \searrow p_c$, and regularity properties for large holes in the infinite cluster. Volume-frozen percolation also gives insight into forest-fire processes, where lightning hits independently each tree with a small rate, and burns its entire connected component immediately.

math.PR↗

arXiv2015

Two-dimensional volume-frozen percolation: exceptional scales

We study a percolation model on the square lattice, where clusters "freeze" (stop growing) as soon as their volume (i.e. the number of sites they contain) gets larger than N, the parameter of the model. A model where clusters freeze when they reach diameter at least N was studied in earlier papers. Using volume as a way to measure the size of a cluster - instead of diameter - leads, for large N, to a quite different behavior (contrary to what happens on the binary tree, where the volume model and the diameter model are "asymptotically the same"). In particular, we show the existence of a sequence of "exceptional" length scales.

math.PR↗

arXiv2013

Embedding binary sequences into Bernoulli site percolation on $\mathbb{Z}^3$

We investigate the problem of embedding infinite binary sequences into Bernoulli site percolation on $\mathbb{Z}^d$ with parameter $p$, known also as percolation of words.\ In 1995, I.\ Benjamini and H.\ Kesten proved that, for $d \geq 10$ and $p=1/2$, all sequences can be embedded, almost surely. They conjectured that the same should hold for $d \geq 3$. In this paper we consider $d \geq 3$ and $p \in (p_c(d), 1-p_c(d))$, where $p_c(d)<1/2$ is the critical threshold for site percolation on $\mathbb{Z}^d$. We show that there exists an integer $M = M (p)$, such that, a.s., every binary sequence, for which every run of consecutive {0s} or {1s} contains at least $M$ digits, can be embedded.

math.PR↗

arXiv2013

Percolation on uniform infinite planar maps

We construct the uniform infinite planar map (UIPM), obtained as the n \to \infty local limit of planar maps with n edges, chosen uniformly at random. We then describe how the UIPM can be sampled using a "peeling" process, in a similar way as for uniform triangulations. This process allows us to prove that for bond and site percolation on the UIPM, the percolation thresholds are p_c^bond=1/2 and p_c^site=2/3 respectively. This method also works for other classes of random infinite planar maps, and we show in particular that for bond percolation on the uniform infinite planar quadrangulation, the percolation threshold is p_c^bond=1/3.

math.PR↗

arXiv2012

On monochromatic arm exponents for 2D critical percolation

We investigate the so-called monochromatic arm exponents for critical percolation in two dimensions. These exponents, describing the probability of observing j disjoint macroscopic paths, are shown to exist and to form a different family from the (now well understood) polychromatic exponents. More specifically, our main result is that the monochromatic j-arm exponent is strictly between the polychromatic j-arm and (j+1)-arm exponents.

math.PR↗