arXiv subjects

Pierre-François Rodriguez

Publications and source records attributed to Pierre-François Rodriguez.

At least 19 recordsLinked to original sources

arXiv2026

Fractal dimension of critical Gaussian free field sign clusters

We study metric graph sign clusters of the Gaussian free field on a large class of transient weighted graphs $G$ that are sufficiently low-dimensional, i.e. below the mean-field regime, and prove a macroscopic equivalence property: several natural ways to define macroscopic clusters - involving any of diameter, volume, or capacity functionals - are in fact the same. As a corollary, macroscopic clusters are characterized in low dimensions as those containing large loops in the corresponding loop soup picture, in contrast with the typical behavior in high dimensions. We then apply these results to the small-mesh scaling limit of critical clusters on $\varepsilon\mathbb{Z}^d$, $d=3,4,5$, as $\varepsilon \downarrow 0$. We prove that every subsequential scaling limit consists of sets with positive Brownian capacity whose Hausdorff and Minkowski dimensions are both equal to $1+\tfrac d2$, as conjectured by Werner. We also derive two-point crossing estimates in the limit, and show that each limiting loop soup cluster is the closure of the union of the scaling limits of the macroscopic loops it contains. Remarkably, unlike with planar Bernoulli percolation for instance, the determination of the Hausdorff dimension does not emerge from direct calculations involving a limiting object (such as $\text{SLE}_6$), but rather from the distinctive features of the discrete model itself.

math.PR↗

arXiv2026

Sharp connectivity bounds for the vacant set of random interlacements

We consider percolation of the vacant set of random interlacements at intensity $u$ in dimensions three and higher, and derive lower bounds on the truncated two-point function for all values of $u>0$. These bounds are sharp up to principal exponential order for all $u$ in dimension three and all $u \neq u_\ast$ in higher dimensions, where $u_*$ refers to the critical parameter of the model, and they match the upper bounds derived in the article arXiv:2503.14497. In dimension three, our results further imply that the truncated two-point function grows at large distances $x$ at a rate that depends on $x$ only through its Euclidean norm, which offers a glimpse of the expected (Euclidean) invariance of the scaling limit at criticality. The rate function is atypical, it incurs a logarithmic correction and comes with an explicit pre-factor that converges to $0$ as the parameter $u$ approaches the critical point $u_*$ from either side. A particular challenge stems from the combined effects of lack of monotonicity due to the truncation in the super-critical phase, and the precise (rotationally invariant) controls we seek, that measure the effects of a certain "harmonic humpback" function. Among others, their derivation relies on rather fine estimates for hitting probabilities of the random walk in arbitrary direction $e$, which witness this invariance at the discrete level, and preclude straightforward applications of projection arguments.

math.PR↗

arXiv2025

Anomalous scaling law for the two-dimensional Gaussian free field

We consider the Gaussian free field $φ$ on $\mathbb{Z}^2$ at large spatial scales $N$ and give sharp bounds on the probability $θ(a,N)$ that the radius of a finite cluster in the excursion set $\{φ\geq a\}$ on the corresponding metric graph is macroscopic. We prove a scaling law for this probability, by which $θ(a,N)$ transitions from fractional logarithmic decay for near-critical parameters $(a,N)$ to polynomial decay in the off-critical regime. The transition occurs across a certain scaling window determined by a correlation length scale $ξ$, which is such that $θ(a,N) \sim θ(0,ξ)(\tfrac{N}ξ)^{-τ}$ for typical heights $a$ as $N/ξ$ diverges, with an explicit exponent $τ$ that we identify in the process. This is in stark contrast with recent results from arXiv:2101.02200 and arXiv:2312.10030 in dimension three, where similar observables are shown to follow regular scaling laws, with polynomial decay at and near criticality, and rapid decay in ${N}/ξ$ away from it.

math.PR↗

arXiv2025

On cable-graph percolation between dimensions 2 and 3

We consider the Gaussian free field on two-dimensional slabs with a thickness described by a height $h$ at spatial scale $N$. We investigate the radius of critical clusters for the associated cable-graph percolation problem, which depends sensitively on the parameter $h$. Our results unveil a whole family of new "fixed points", which interpolate between recent results from arXiv:2303.03782 in two dimensions and from arXiv:2405.17417 and arXiv:2406.02397 in three dimensions, and describe critical behaviour beyond those regimes. In the delocalised phase, the one-arm decay exhibits a "plateau", i.e. it doesn't depend on the speed at which the variance of the field diverges in the large-$N$ limit. Our methods rely on a careful analysis of the interplay between two- and three-dimensional effects for the underlying random walk, which manifest themselves in a corresponding decomposition of the field.

math.PR↗

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↗

arXiv2025

Strong local uniqueness for the vacant set of random interlacements

We consider the the vacant set $\mathcal{V}^u$ of random interlacements on $\mathbb{Z}^d$ in dimensions $d \ge 3$. For varying intensity $u > 0$, the connectivity properties of $\mathcal V^u$ undergo a percolation phase transition across a critical parameter $u_* \in (0,\infty)$. In this article, we prove that this phase transition is sharp in the supercritical phase $u < u_*$. This follows from a certain strong local uniqueness property (SLU) introduced in the present work, which we prove $\mathcal{V}^u$ satisfies. In itself, this property furnishes the missing ingredient needed to deduce a number of desirable quenched results characterizing the large-scale geometry of the infinite cluster. Moreover, SLU entails a sought-after local and monotone criterion amenable to renormalization arguments below $u_*$.

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

Sharp threshold for the ballisticity of the random walk on the exclusion process

We study a non-reversible random walk advected by the symmetric simple exclusion process, so that the walk has a local drift of opposite sign when sitting atop an occupied or an empty site. We prove that the back-tracking probability of the walk exhibits a sharp transition as the density $ρ$ of particles in the underlying exclusion process varies across a critical density $ρ_c$. Our results imply that the speed $v=v(ρ)$ of the walk is a strictly monotone function and that the zero-speed regime is either absent or collapses to a single point, $ρ_c$, thus solving a conjecture of arXiv:1906.03167. The proof proceeds by exhibiting a quantitative monotonicity result for the speed of a truncated model, in which the environment is renewed after a finite time horizon $L$. The truncation parameter $L$ is subsequently pitted against the density $ρ$ to carry estimates over to the full model. Our strategy is somewhat reminiscent of certain techniques recently used to prove sharpness results in percolation problems. A key instrument is a combination of renormalisation arguments with refined couplings of environments at slightly different densities, which we develop in this article. Our results hold in fact in greater generality and apply to a class of environments with possibly egregious features, outside perturbative regimes.

math.PR↗

arXiv2024

Equality of critical parameters for percolation of Gaussian free field level-sets

We consider upper level-sets of the Gaussian free field on $\mathbb Z^d$, for $d\geq 3$, above a given real-valued height parameter $h$. As $h$ varies, this defines a canonical percolation model with strong, algebraically decaying correlations. We prove that three natural critical parameters associated to this model, respectively describing a well-ordered subcritical phase, the emergence of an infinite cluster, and the onset of a local uniqueness regime in the supercritical phase, actually coincide. At the core of our proof lies a new interpolation scheme aimed at integrating out the long-range dependence of the Gaussian free field. Due to the strength of correlations, its successful implementation requires that we work in an effectively critical regime. Our analysis relies extensively on certain novel renormalization techniques that bring into play all relevant scales simultaneously. The approach in this article paves the way to a complete understanding of the off-critical phases for strongly correlated disordered systems.

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

A Ray-Knight theorem for $\nablaϕ$ interface models and scaling limits

We introduce a natural measure on bi-infinite random walk trajectories evolving in a time-dependent environment driven by the Langevin dynamics associated to a gradient Gibbs measure with convex potential. We derive an identity relating the occupation times of the Poissonian cloud induced by this measure to the square of the corresponding gradient field, which - generically - is not Gaussian. In the quadratic case, we recover a well-known generalization of the second Ray-Knight theorem. We further determine the scaling limits of the various objects involved in dimension 3, which are seen to exhibit homogenization. In particular, we prove that the renormalized square of the gradient field converges under appropriate rescaling to the Wick-ordered square of a Gaussian free field on $\mathbb{R}^3$ with suitable diffusion matrix, thus extending a celebrated result of Naddaf and Spencer regarding the scaling limit of the field itself.

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

The Discrete Gaussian model, I. Renormalisation group flow at high temperature

The Discrete Gaussian model is the lattice Gaussian free field conditioned to be integer-valued. In two dimensions, at sufficiently high temperature, we show that its macroscopic scaling limit on the torus is a multiple of the Gaussian free field. Our proof starts from a single renormalisation group step after which the integer-valued field becomes a smooth field which we then analyse using the renormalisation group method. This paper also provides the foundation for the construction of the scaling limit of the infinite-volume gradient Gibbs state of the Discrete Gaussian model in the companion paper. Moreover, we develop all estimates for general finite-range interaction with sharp dependence on the range. We expect these estimates to prepare for a future analysis of the spread-out version of the Discrete Gaussian model at its critical temperature.

math.PR↗

arXiv2023

The Discrete Gaussian model, II. Infinite-volume scaling limit at high temperature

The Discrete Gaussian model is the lattice Gaussian free field conditioned to be integer-valued. In two dimensions, at sufficiently high temperature, we show that the scaling limit of the infinite-volume gradient Gibbs state with zero mean is a multiple of the Gaussian free field. This article is the second in a series on the Discrete Gaussian model, extending the methods of the first paper by the analysis of general external fields (rather than macroscopic test functions on the torus). As a byproduct, we also obtain a scaling limit for mesoscopic test functions on the torus.

math.PR↗

arXiv2023

Phase transition for the vacant set of random walk and random interlacements

We consider the set of points visited by the random walk on the discrete torus $(\mathbb{Z}/N\mathbb{Z})^d$, for $d \geq 3$, at times of order $uN^d$, for a parameter $u>0$ in the large-$N$ limit. We prove that the vacant set left by the walk undergoes a phase transition across a non-degenerate critical value $u_* = u_*(d)$, as follows. For all $u< u_*$, the vacant set contains a giant connected component with high probability, which has a non-vanishing asymptotic density and satisfies a certain local uniqueness property. In stark contrast, for all $u> u_*$ the vacant set scatters into tiny connected components. Our results further imply that the threshold $u_*$ precisely equals the critical value, introduced by Sznitman in arXiv:0704.2560, which characterizes the percolation transition of the corresponding local limit, the vacant set of random interlacements on $\mathbb{Z}^d$. Our findings also yield the analogous infinite-volume result, i.e. the long purported equality of three critical parameters $\bar u$, $u_*$ and $u_{**}$ naturally associated to the vacant set of random interlacements.

math.PR↗

arXiv2023

A characterization of strong percolation via disconnection

We consider a percolation model, the vacant set $\mathcal{V}^u$ of random interlacements on $\mathbb{Z}^d$, $d \geq 3$, in the regime of parameters $u>0$ in which it is strongly percolative. By definition, such values of $u$ pinpoint a robust subset of the super-critical phase, with strong quantitative controls on large local clusters. In the present work, we give a new charaterization of this regime in terms of a single property, monotone in $u$, involving a disconnection estimate for $\mathcal{V}^u$. A key aspect is to exhibit a gluing property for large local clusters from this information alone, and a major challenge in this undertaking is the fact that the conditional law of $\mathcal{V}^u$ exhibits degeneracies. As one of the main novelties of this work, the gluing technique we develop to merge large clusters accounts for such effects. In particular, our methods do not rely on the widely assumed finite-energy property, which the set $\mathcal{V}^u$ does not possess. The charaterization we derive plays a decisive role in the proof of a lasting conjecture regarding the coincidence of various critical parameters naturally associated to $\mathcal{V}^u$ in a companion article.

math.PR↗