arXiv subjects

David Dereudre

Publications and source records attributed to David Dereudre.

At least 19 recordsLinked to original sources

arXiv2026

First-order phase transition for Gibbs point processes with saturated interactions

We study first-order phase transitions in continuum Gibbs point processes with saturated interactions. These interactions form a broad class of Hamiltonians in which the local energy in regions of high particle density depends only on the number of points. Building on ideas of Pirogov-Sinai-Zahradnik theory and its adaptations to the continuum, we develop a general method for establishing the existence of two distinct infinite-volume Gibbs measures with different intensities in this setting, demonstrating a first-order phase transition. Our approach extends previous results obtained for the Quermass model and applies in particular to a new class of diluted pairwise interactions introduced in this work.

math.PR↗

arXiv2025

Rigidity of one-dimensional point processes via optimal transport

We investigate rigidity phenomena in one-dimensional point processes. We show that the existence of an $L^1$ transport map from a stationary lattice or the Lebesgue measure to a point process is sufficient to guarantee the properties of Number-Rigidity and Cyclic-Factor. We then apply this result to non-singular Riesz gases with parameter $s\in(-2,-1]$, defined in infinite volume as accumulation points of stationarized finite-volume Riesz gases. This includes, for $s=-1$, the well-known one-dimensional Coulomb gas (also called Jellium plasma, or the one-component 1D plasma).

math.PR↗

arXiv2024

DLR Equations for the Superstable Bose Gas at any Temperature and Activity

We construct a thermodynamic limit for the grand canonical Bose gas in dimension $d\geqslant1$ (in its Feynman-Kac representation) with superstable interaction at any inverse temperature $\beta>0$ and any chemical potential $\mu\in\mathbb{R}$. Our infinite volume model is naturally a distribution over configurations of finite loops and possibly interlacements. We prove the limiting process to solve a new class of DLR equations involving random permutations and Brownian paths.

math-ph↗

arXiv2024

Absence of percolation for infinite Poissonian systems of stopped paths

The state space of our model is the Euclidean space in dimension d = 2. Simultaneously, from all points of a homogeneous Poisson point process, we let grow independent and identically distributed random continuum paths. Each path stops growing at time t \> 0 if it hits the trace of the other curves realized up until time t. Such dynamic is well-defined as long as the distribution of paths has a finite second moment at each time t \> 0. Letting the time runs until infinity so that each path reaches its stopping curve, we study the connected property of the graph formed by all stopped curves. Our main result states the absence of percolation in this graph, meaning that each cluster consists of a finite number of curves. The assumptions on the distribution of paths are very mild, with the main one being the so-called 'loop assumption' which ensures that finite clusters (necessarily containing a loop) occur with positive probability. The main issue in this model comes from the long-range dependence arising from long sequences of causalities in the hitting/stopping procedure. Most methods based on block approaches fail to effectively address the question of percolation in this setting.

math.PR↗

arXiv2024

(Non)-hyperuniformity of perturbed lattices

We ask whether a stationary lattice in dimension $d$ whose points are shifted by identically distributed but possibly dependent perturbations remains hyperuniform. When $d = 1$ or $2$, we show that it is the case when the perturbations have a finite $d$-moment, and that this condition is sharp. When $d \geq 3$, we construct arbitrarily small perturbations such that the resulting point process is not hyperuniform. As a side remark of independent interest, we exhibit hyperuniform processes with arbitrarily slow decay of their number variance.

math.PR↗

arXiv2024

Non-hyperuniformity of Gibbs point processes with short range interaction

We investigate the hyperuniformity of marked Gibbs point processes with weak dependencies among distant points whilst the interactions of close points are kept arbitrary. Some variants of stability and range assumptions are posed on the Papangelou intensity in order to prove that the resulting point process is not hyperuniform. The scope of our results covers many frequently used models including Gibbs point processes with a superstable, lower-regular, integrable pair potential as well as Widom--Rowlinson model with random radii or Gibbs point processes with interactions based on Voronoi tessellation and nearest neighbour graph.

math.PR↗

arXiv2023

Liquid-Gas phase transition for Gibbs point process with Quermass interaction

We prove the existence of a liquid-gas phase transition for continuous Gibbs point process in $\mathbb{R}^d$ with Quermass interaction. The Hamiltonian we consider is a linear combination of the volume $\mathcal{V}$, the surface measure $\mathcal{S}$ and the Euler-Poincar\'e characteristic $\chi$ of a halo of particles (i.e. an union of balls centred at the positions of particles). We show the non-uniqueness of infinite volume Gibbs measures for special values of activity and temperature, provided that the temperature is low enough. Moreover we show the non-differentiability of the pressure at these critical points. Our main tool is an adaptation of the Pirogov-Sina\"i-Zahradnik theory for continuous systems with interaction exhibiting a saturation property.

math.PR↗

arXiv2021

Number-Rigidity and $β$-Circular Riesz gas

For an inverse temperature $β>0$, we define the $β$-circular Riesz gas on $\mathbb{R}^d$ as any microscopic thermodynamic limit of Gibbs particle systems on the torus interacting via the Riesz potential $g(x) = \Vert x \Vert^{-s}$. We focus on the non integrable case $d-1 0$, the existence of a $β$-circular Riesz gas which is not number-rigid. Recall that a point process is said number rigid if the number of points in a bounded Borel set $Δ$ is a function of the point configuration outside $Δ$. It is the first time that the non number-rigidity is proved for a Gibbs point process interacting via a non integrable potential. We follow a statistical physics approach based on the canonical DLR equations. It is inspired by Dereudre-Hardy-Leblé and Maïda (2021) where the authors prove the number-rigidity of the $\text{Sine}_β$ process.

math.PR↗

arXiv2021

Fully-connected bond percolation on $\mathbb{Z}^d$

We consider the bond percolation model on the lattice $\mathbb{Z}^d$ ($d\ge 2$) with the constraint to be fully connected. Each edge is open with probability $p\in(0,1)$, closed with probability $1-p$ and then the process is conditioned to have a unique open connected component (bounded or unbounded). The model is defined on $\mathbb{Z}^d$ by passing to the limit for a sequence of finite volume models with general boundary conditions. Several questions and problems are investigated: existence, uniqueness, phase transition, DLR equations. Our main result involves the existence of a threshold $0<p^*(d)<1$ such that any infinite volume process is necessary the vacuum state in subcritical regime (no open edges) and is non trivial in the supercritical regime (existence of a stationary unbounded connected cluster). Bounds for $p^*(d)$ are given and show that it is drastically smaller than the standard bond percolation threshold in $\mathbb{Z}^d$. For instance $0.128<p^*(2)<0.202$ (rigorous bounds) whereas the 2D bond percolation threshold is equal to $1/2$.

math.PR↗

arXiv2020

Existence and percolation results for stopped germ-grain models with unbounded velocities

We investigate the existence and first percolation properties of general stopped germ-grain models. They are defined via a random set of germs generated by a homogeneous planar Poisson point process in $\mathbf{R}^{2}$. From each germ, a grain, composed by a random number of branches, grows. This grain stops to grow whenever one of its branches hits another grain. The classical and historical example is the line segment model for which the grains are segments growing in a random direction in $ [0,2π)$ with random velocity. In the bilateral line segment model the segments grow in both directions. Other examples are considered here such as the Brownian model where the branches are simply given by independent Brownian motions in $\mathbf{R}^{2}$. The existence of such dynamics for an infinite number of germs is not obvious and our first result ensures it in a very general setting. In particular the existence of the line segment model is proved as soon as the random velocity admits a moment of order 4 which extends the result by Daley et al (Theorem 4.3 in \cite{daley2014two}) for bounded velocity. Our result covers also the Brownian dynamic model. In a second part of the paper, we show that the line segment model with random velocity admitting a super exponential moment does not percolate. This improves a recent result (Theorem 3.2 \cite{coupier2016absence}) in the case of bounded velocity.

math.PR↗

arXiv2020

Sharp phase transition for the continuum Widom-Rowlinson model

The Widom-Rowlinson model (or the Area-interaction model) is a Gibbs point process in $\mathbb{R}^d$ with the formal Hamiltonian $H(ω)=\text{Volume}(\cup_{x\inω} B_1(x))$, where $ω$ is a locally finite configuration of points and $B_1(x)$ denotes the unit closed ball centred at $x$. The model is tuned by two parameters: the activity $z>0$ and the inverse temperature $β\ge 0$. We investigate the phase transition of the model in the point of view of percolation theory and the liquid-gas transition. First, considering the graph connecting points with distance smaller than $2r>0$, we show that for any $β>0$, there exists $0<\tilde{z}^a(β, r)<+\infty$ such that an exponential decay of connectivity at distance $n$ occurs in the subcritical phase and a linear lower bound of the connection at infinity holds in the supercritical case. Secondly we study a standard liquid-gas phase transition related to the uniqueness/non-uniqueness of Gibbs states depending on the parameters $z,β$. Old results claim that a non-uniqueness regime occurs for $z=β$ large enough and it is conjectured that the uniqueness should hold outside such an half line ($z=β\ge β_c>0$). We solve partially this conjecture by showing that for $β$ large enough the non-uniqueness holds if and only if $z=β$. We show also that this critical value $z=β$ corresponds to the percolation threshold $ \tilde{z}^a(β, r)=β$ for $β$ large enough, providing a straight connection between these two notions of phase transition.

math.PR↗

arXiv2019

DLR equations and rigidity for the Sine-beta process

We investigate Sine$_β$, the universal point process arising as the thermodynamic limit of the microscopic scale behavior in the bulk of one-dimensional log-gases, or $β$-ensembles, at inverse temperature $β>0$. We adopt a statistical physics perspective, and give a description of Sine$_β$ using the Dobrushin-Lanford-Ruelle (DLR) formalism by proving that it satisfies the DLR equations: the restriction of Sine$_β$ to a compact set, conditionally to the exterior configuration, reads as a Gibbs measure given by a finite log-gas in a potential generated by the exterior configuration. Moreover, we show that Sine$_β$ is number-rigid and tolerant in the sense of Ghosh-Peres, i.e. the number, but not the position, of particles lying inside a compact set is a deterministic function of the exterior configuration. Our proof of the rigidity differs from the usual strategy and is robust enough to include more general long range interactions in arbitrary dimension.

math.PR↗

arXiv2019

Absence of percolation for Poisson outdegree-one graphs

A Poisson outdegree-one graph is an oriented graph based on a Poisson point process such that each vertex has only one outgoing edge. The paper focuses on the absence of percolation for such graphs. Our main result is based on two assumptions. The Shield assumption ensures that the graph is locally determined with possible random horizons. The Loop assumption ensures that any forward branch of the graph merges on a loop provided that the Poisson point process is augmented with a finite collection of well-chosen points. Several models satisfy these general assumptions and inherit in consequence the absence of percolation. In particular, we solve a conjecture by Daley et al. on the absence of percolation for the line-segment model. In this planar model, a segment is growing from any point of the Poisson process and stops its growth whenever it hits another segment. The random directions are picked independently and uniformly on the unit sphere. Another model of geometric navigation is presented and also fulfills the Shield and Loop assumptions.

math.PR↗

arXiv2019

Existence of Gibbs point processes with stable infinite range interaction

We provide a new proof of the existence of Gibbs point processes with infinite range interactions, based on the compactness of entropy levels. Our main existence theorem holds under two assumptions. The first one is the standard stability assumption, which means that the energy of any finite configuration is super-linear with respect to the number of points. The second assumption is the so-called intensity regularity, which controls the long range of the interaction via the intensity of the process. This assumption is new and introduced here since it is well adapted to the entropy approach. As a corollary of our main result we improve the existence results by Ruelle for pairwise interactions by relaxing the superstabilty assumption. Note that our setting is not reduced to pairwise interaction and can contain infinite range multi-body counterparts.

math.PR↗

arXiv2018

Phase transition for continuum Widom-Rowlinson model with random radii

In this paper we study the phase transition of continuum Widom-Rowlinson measures in $\mathbb{R}^d$ with $q$ types of particles and random radii. Each particle $x_i$ of type $i$ is marked by a random radius $r_i$ distributed by a probability measure $Q_i$ on $\mathbb{R}^+$. The particles of same type do not interact each other whereas particles $x_i$ and $x_j$ with different type $i \neq j$ interact via an exclusion hardcore interaction forcing $r_i+r_j$ to be smaller than $|x_i-x_j|$. In the integrable case (i.e. $\int r^d Q_i(dr)<+\infty$, $1\le i\le q$), we show that the Widom-Rowlinson measures exhibit a standard phase transition providing uniqueness, when the activity is small, and co-existence of $q$ ordered phases, when the activity is large. In the non-integrable case (i.e. $\int r^d Q_i(dr)=+\infty$, $1\le i \le q$), we show another type of phase transition. We prove, when the activity is small, the existence of at least $q+1$ extremal phases and we conjecture that, when the activity is large, only the $q$ ordered phases subsist. We prove a weak version of this conjecture by showing that the symmetric Widom-Rowlinson measure with free boundary condition is a mixing of the $q$ ordered phases if and only if the activity is large.

math.PR↗

arXiv2018

Introduction to the theory of Gibbs point processes

The Gibbs point processes (GPP) constitute a large class of point processes with interaction between the points. The interaction can be attractive, repulsive, depending on geometrical features whereas the null interaction is associated to the so-called Poisson point process. In a first part of this mini-course, we present several aspects of finite volume GPP defined on a bounded window in Rd. In a second part, we introduce the more complicated formalism of infinite volume GPP defined on the full space Rd. Existence, uniqueness and non-uniqueness of GPP are non-trivial questions which we treat here with completely self-contained proofs. The DLR equations, the GNZ equations and the variational principle are presented as well. Finally, in a last part, we investigate the estimation of parameters. The main standard estimators (MLE, MPLE, Takac-Fiksel and variational estimators) are presented and we prove their consistency. For sake of simplicity, during all the mini-course, we consider only the case of finite range interaction and the setting of marked points is not presented.

math.PR↗

arXiv2017

On the critical threshold for continuum AB percolation

Consider a bipartite random geometric graph on the union of two independent homogeneous Poisson point processes in $d$-space, with distance parameter $r$ and intensities $λ,μ$. For any $λ>0$ we consider the percolation threshold $μ_c(λ)$ associated to the parameter $μ$. Denoting by $λ_c:= λ_c(2r)$ the percolation threshold for the standard Poisson Boolean model with radii $r$, we show the lower bound $μ_c(λ)\ge c\log(c/(λ-λ_c))$ for any $λ>λ_c$ with $c>0$ a fixed constant. In particular, $μ_c(λ)$ tends to infinity when $λ$ tends to $λ_c$ from above.

math.PR↗

arXiv2016

Exact simulation of Brownian diffusions with drift admitting jumps

In this paper, using an algorithm based on the retrospective rejection sampling scheme, we propose an exact simulation of a Brownian diffusion whose drift admits several jumps. We treat explicitly and extensively the case of two jumps, providing numerical simulations. Our main contribution is to manage the technical diffculty due to the presence of two jumps thanks to a new explicit expression of the transition density of the skew Brownian motion with two semipermeable barriers and a constant drift.

math.PR↗