arXiv subjects
Publications and source records attributed to F. den Hollander.
The present paper is a brief overview of random opinion dynamics on random graphs based on the Ising Lecture given by the author at the World Congress in Probability and Statistics, 12--16 August 2024, Bochum, Germany. The content is a snapshot of an interesting area of research that is developing rapidly.
The present overview of interacting particle systems on random graphs collects the notes of a mini-course given by the authors at the Brazilian School of Probability, 5--9 August 2024, in Salvador, Bahia, Brazil. The content is a personal snapshot of an interesting area of research at the interface between probability theory, combinatorics, statistical physics and network science that is developing rapidly.
In [18] we analysed a simple undirected random graph subject to constraints on the total number of edges and the total number of triangles. We considered the dense regime in which the number of edges per vertex is proportional to the number of vertices. We showed that, as soon as the constraints are \emph{frustrated}, i.e., do not lie on the Erdős-Rényi line, there is breaking of ensemble equivalence, in the sense that the specific relative entropy per edge of the \emph{microcanonical ensemble} with respect to the \emph{canonical ensemble} is strictly positive in the limit as the number of vertices tends to infinity. In the present paper we analyse what happens near the Erdős-Rényi line. It turns out that the way in which the specific relative entropy tends to zero depends on whether the total number of triangles is slightly larger or slightly smaller than typical. We investigate what the constrained random graph looks like asymptotically in the microcanonical ensemble.
In this paper we consider a random graph on which topological restrictions are imposed, such as constraints on the total number of edges, wedges, and triangles. We work in the dense regime, in which the number of edges per vertex scales proportionally to the number of vertices $n$. Our goal is to compare the micro-canonical ensemble (in which the constraints are satisfied for every realisation of the graph) with the canonical ensemble (in which the constraints are satisfied on average), both subject to maximal entropy. We compute the relative entropy of the two ensembles in the limit as $n$ grows large, where two ensembles are said to be \emph{equivalent} in the dense regime if this relative entropy divided by $n^2$ tends to zero. Our main result, whose proof relies on large deviation theory for graphons, is that breaking of ensemble equivalence occurs when the constraints are \emph{frustrated}. Examples are provided for three different choices of constraints.
In previous work by Avena and den Hollander, a model of a one-dimensional random walk in a dynamic random environment was proposed where the random environment is resampled from a given law along a growing sequence of deterministic times. In the regime where the increments of the resampling times diverge, which is referred to as the cooling regime, a weak law of large numbers and certain fluctuation properties were derived under the annealed measure. In the present paper we show that a strong law of large numbers and a quenched large deviation principle hold as well. In the cooling regime, the random walk can be represented as a sum of independent variables, distributed as the increments of a random walk in a static random environment over increasing periods of time. Our proofs require suitable multi-layer decompositions of sums of random variables controlled by moments bounds and concentration estimates. Along the way we derive two results of independent interest, namely, a concentration inequality for the cumulants of the displacement in the static random environment and an ergodic theorem that deals with limits of sums of triangular arrays representing the structure of the cooling regime. We close by discussing our present understanding of homogenisation effects as a function of the speed of divergence of the increments of the resampling times.
We obtain bounds for the expected loss of torsional rigidity of a cylinder $Ω_L=(-L/2,L/2) \times Ω\subset \R^3$ of length $L$ due to a Brownian fracture that starts at a random point in $Ω_L,$ and runs until the first time it exits $Ω_L$. These bounds are expressed in terms of the geometry of the cross-section $Ω\subset \R^2$. It is shown that if $Ω$ is a disc with radius $R$, then in the limit as $L \rightarrow \infty$ the expected loss of torsional rigidity equals $cR^5$ for some $c\in (0,\infty)$. We derive bounds for $c$ in terms of the expected Newtonian capacity of the trace of a Brownian path that starts at the centre of a ball in $\R^3$ with radius $1,$ and runs until the first time it exits this ball.
We propose a model of a one-dimensional random walk in dynamic random environment that interpolates between two classical settings: (I) the random environment is sampled at time zero only; (II) the random environment is resampled at every unit of time. In our model the random environment is resampled along an increasing sequence of deterministic times. We consider the annealed version of the model, and look at three growth regimes for the resampling times: (R1) linear; (R2) polynomial; (R3) exponential. We prove weak laws of large numbers and central limit theorems. We list some open problems and conjecture the presence of a crossover for the scaling behaviour in regimes (R2) and (R3).
In this paper we prove a version of the Berman-Konsowa principle for reversible Markov jump processes on Polish spaces. The Berman-Konsowa principle provides a variational formula for the capacity of a pair of disjoint measurable sets. There are two versions, one involving a class of probability measures for random finite paths from one set to the other, the other involving a class of finite unit flows from one set to the other. The Berman-Konsowa principle complements the Dirichlet principle and the Thomson principle, and turns out to be especially useful for obtaining sharp estimates on crossover times in metastable interacting particle systems.
In this paper we consider $β[0; s]$, Brownian motion of time length $s > 0$, in $m$-dimensional Euclidean space $\mathbb R^m$ and on the $m$-dimensional torus $\mathbb T^m$. We compute the expectation of (i) the heat content at time $t$ of $\mathbb R^m\setminus β[0; s]$ for fixed $s$ and $m = 2,3$ in the limit $t \downarrow 0$, when $β[0; s]$ is kept at temperature 1 for all $t > 0$ and $\mathbb R^m\setminus β[0; s]$ has initial temperature 0, and (ii) the inradius of $\mathbb R^m\setminus β[0; s]$ for $m = 2,3,\cdots$ in the limit $s \rightarrow \infty$.
In this paper we consider a two-dimensional copolymer consisting of a random concatenation of hydrophobic and hydrophilic monomers near a linear interface separating oil and water acting as solvents. The configurations of the copolymer are directed paths that can move above and below the interface. The interaction Hamiltonian, which rewards matches and penalizes mismatches of the monomers and the solvents, depends on two parameters: the interaction strength $β\geq 0$ and the interaction bias $h \geq 0$. The quenched excess free energy per monomer $(β,h) \mapsto g^\mathrm{que} (β,h)$ has a phase transition along a quenched critical curve $β\mapsto h^\mathrm{que}_c(β)$ separating a localized phase, where the copolymer stays close to the interface, from a delocalized phase, where the copolymer wanders away from the interface. We derive a variational expression for $g^\mathrm{que}(β,h)$ by applying the quenched large deviation principle for the empirical process of words cut out from a random letter sequence according to a random renewal process. We compare this variational expression with its annealed analogue, describing the annealed excess free energy $(β,h) \mapsto g^\mathrm{ann}(β,h)$, which has a phase transition along an annealed critical curve $β\mapsto h^\mathrm{ann}_c(β)$. Our results extend to a general class of disorder distributions and directed paths. We show that $g^\mathrm{que}(β,h) 0$ when $α>1$. This gap vanished when $α=1$.
This is the first in a series of three papers in which we study a two-dimensional lattice gas consisting of two types of particles subject to Kawasaki dynamics at low temperature in a large finite box with an open boundary. Each pair of particles occupying neighboring sites has a negative binding energy provided their types are different, while each particle has a positive activation energy that depends on its type. There is no binding energy between neighboring particles of the same type. At the boundary of the box particles are created and annihilated in a way that represents the presence of an infinite gas reservoir. We start the dynamics from the empty box and compute the transition time to the full box. This transition is triggered by a \emph{critical droplet} appearing somewhere in the box. We identify the region of parameters for which the system is metastable. For this region, in the limit as the temperature tends to zero, we show that the first entrance distribution on the set of critical droplets is uniform, compute the expected transition time up to a multiplicative factor that tends to one, and prove that the transition time divided by its expectation is exponentially distributed. These results are derived under \emph{three hypotheses} on the energy landscape, which are verified in the second and the third paper for a certain subregion of the metastable region. These hypotheses involve three model-dependent quantities -- the energy, the shape and the number of the critical droplets -- which are identified in the second and the third paper as well.
In this paper we study intermittency for the parabolic Anderson equation $\partial u/\partial t=κΔu+γξu$ with $u:\mathbb{Z}^d\times[0,\infty)\to\mathbb{R}$, where $κ\in[0,\infty)$ is the diffusion constant, $Δ$ is the discrete Laplacian, $γ\in(0,\infty)$ is the coupling constant, and $ξ:\mathbb{Z}^d\times[0,\infty)\to\mathbb{R}$ is a space--time random medium. The solution of this equation describes the evolution of a ``reactant'' $u$ under the influence of a ``catalyst'' $ξ$. We focus on the case where $ξ$ is the voter model with opinions 0 and 1 that are updated according to a random walk transition kernel, starting from either the Bernoulli measure $ν_ρ$ or the equilibrium measure $μ_ρ$, where $ρ\in(0,1)$ is the density of 1's. We consider the annealed Lyapunov exponents, that is, the exponential growth rates of the successive moments of $u$. We show that if the random walk transition kernel has zero mean and finite variance, then these exponents are trivial for $1\leq d\leq4$, but display an interesting dependence on the diffusion constant $κ$ for $d\geq 5$, with qualitatively different behavior in different dimensions. In earlier work we considered the case where $ξ$ is a field of independent simple random walks in a Poisson equilibrium, respectively, a symmetric exclusion process in a Bernoulli equilibrium, which are both reversible dynamics. In the present work a main obstacle is the nonreversibility of the voter model dynamics, since this precludes the application of spectral techniques. The duality with coalescing random walks is key to our analysis, and leads to a representation formula for the Lyapunov exponents that allows for the application of large deviation estimates.
Consider a one-dimensional shift-invariant attractive spin-flip system in equilibrium, constituting a dynamic random environment, together with a nearest-neighbor random walk that on occupied sites has a local drift to the right but on vacant sites has a local drift to the left. In previous work we proved a law of large numbers for dynamic random environments satisfying a space-time mixing property called cone-mixing. If an attractive spin-flip system has a finite average coupling time at the origin for two copies starting from the all-occupied and the all-vacant configuration, respectively, then it is cone-mixing. In the present paper we prove a large deviation principle for the empirical speed of the random walk, both quenched and annealed, and exhibit some properties of the associated rate functions. Under an exponential space-time mixing condition for the spin-flip system, which is stronger than cone-mixing, the two rate functions have a unique zero, i.e., the slow-down phenomenon known to be possible in a static random environment does not survive in a fast mixing dynamic random environment. In contrast, we show that for the simple symmetric exclusion dynamics, which is not cone-mixing (and which is not a spin-flip system either), slow-down does occur.
In this paper we consider a class of one-dimensional interacting particle systems in equilibrium, constituting a dynamic random environment, together with a nearest-neighbor random walk that on occupied/vacant sites has a local drift to the right/left. We adapt a regeneration-time argument originally developed by Comets and Zeitouni for static random environments to prove that, under a space-time mixing property for the dynamic random environment called cone-mixing, the random walk has an a.s. constant global speed. In addition, we show that if the dynamic random environment is exponentially mixing in space-time and the local drifts are small, then the global speed can be written as a power series in the size of the local drifts. From the first term in this series the sign of the global speed can be read off. The results can be easily extended to higher dimensions.
We continue our study of intermittency for the parabolic Anderson model $\partial u/\partial t = κΔu + ξu$ in a space-time random medium $ξ$, where $κ$ is a positive diffusion constant, $Δ$ is the lattice Laplacian on $\Z^d$, $d \geq 1$, and $ξ$ is a simple symmetric exclusion process on $\Z^d$ in Bernoulli equilibrium. This model describes the evolution of a \emph{reactant} $u$ under the influence of a \emph{catalyst} $ξ$. In Gärtner, den Hollander and Maillard (2007) we investigated the behavior of the annealed Lyapunov exponents, i.e., the exponential growth rates as $t\to\infty$ of the successive moments of the solution $u$. This led to an almost complete picture of intermittency as a function of $d$ and $κ$. In the present paper we finish our study by focussing on the asymptotics of the Lyaponov exponents as $κ\to\infty$ in the \emph{critical} dimension $d=3$, which was left open in Gärtner, den Hollander and Maillard (2007) and which is the most challenging. We show that, interestingly, this asymptotics is characterized not only by a \emph{Green} term, as in $d\geq 4$, but also by a \emph{polaron} term. The presence of the latter implies intermittency of \emph{all} orders above a finite threshold for $κ$.
Let $(\{X_i(t)\}_{i\in \mathbb{Z}^d})_{t\geq 0}$ be the system of interacting diffusions on $[0,\infty)$ defined by the following collection of coupled stochastic differential equations: \begin{eqnarray}dX_i(t)=\sum\limits_{j\in \mathbb{Z}^d}a(i,j)[X_j(t)-X_i(t)] dt+\sqrt{bX_i(t)^2} dW_i(t), \eqntext{i\in \mathbb{Z}^d,t\geq 0.}\end{eqnarray} Here, $a(\cdot,\cdot)$ is an irreducible random walk transition kernel on $\mathbb{Z}^d\times \mathbb{Z}^d$, $b\in (0,\infty)$ is a diffusion parameter, and $(\{W_i(t)\}_{i\in \mathbb{Z}^d})_{t\geq 0}$ is a collection of independent standard Brownian motions on $\mathbb{R}$. The initial condition is chosen such that $\{X_i(0)\}_{i\in \mathbb{Z}^d}$ is a shift-invariant and shift-ergodic random field on $[0,\infty)$ with mean $Θ\in (0,\infty)$ (the evolution preserves the mean). We show that the long-time behavior of this system is the result of a delicate interplay between $a(\cdot,\cdot)$ and $b$, in contrast to systems where the diffusion function is subquadratic. In particular, let $\hat{a}(i,j)={1/2}[a(i,j)+a(j,i)]$, $i,j\in \mathbb{Z}^d$, denote the symmetrized transition kernel. We show that: (A) If $\hat{a}(\cdot,\cdot)$ is recurrent, then for any $b>0$ the system locally dies out. (B) If $\hat{a}(\cdot,\cdot)$ is transient, then there exist $b_*\geq b_2>0$ such that: (B1)d The system converges to an equilibrium $ν_Θ$ (with mean $Θ$) if $0 b_*$. (B3) $ν_Θ$ has a finite 2nd moment if and only if $0 b_2$. The equilibrium $ν_Θ$ is shown to be associated and mixing for all $0 b_2$. We further conjecture that the system locally dies out at $b=b_*$. For the case where $a(\cdot,\cdot)$ is symmetric and transient we further show that: (C) There exists a sequence $b_2\geq b_3\geq b_4\geq ... >0$ such that: (C1) $ν_Θ$ has a finite $m$th moment if and only if $0 b_m$. (C3) $b_2\leq (m-1)b_m<2$. uad(C4) $\lim_{m\to\infty}(m-1)b_m=c=\sup_{m\geq 2}(m-1)b_m$. The proof of these results is based on self-duality and on a representation formula through which the moments of the components are related to exponential moments of the collision local time of random walks. Via large deviation theory, the latter lead to variational expressions for $b_*$ and the $b_m$'s, from which sharp bounds are deduced. The critical value $b_*$ arises from a stochastic representation of the Palm distribution of the system. The special case where $a(\cdot,\cdot)$ is simple random walk is commonly referred to as the parabolic Anderson model with Brownian noise. This case was studied in the memoir by Carmona and Molchanov [Parabolic Anderson Problem and Intermittency (1994) Amer. Math. Soc., Providence, RI], where part of our results were already established.
In this paper we review some recent results, obtained jointly with Stu Whittington, for a mathematical model describing a copolymer in an emulsion. The copolymer consists of hydrophobic and hydrophilic monomers, concatenated randomly with equal density. The emulsion consists of large blocks of oil and water, arranged in a percolation-type fashion. To make the model mathematically tractable, the copolymer is allowed to enter and exit a neighboring pair of blocks only at diagonally opposite corners. The energy of the copolymer in the emulsion is minus $α$ times the number of hydrophobic monomers in oil minus $β$ times the number of hydrophilic monomers in water. Without loss of generality we may assume that the interaction parameters are restricted to the cone $\{(α,β)\in \mathbb{R}^2\colon |β|\leqα\}$. We show that the phase diagram has two regimes: (1) in the supercritical regime where the oil blocks percolate, there is a single critical curve in the cone separating a localized and a delocalized phase; (2) in the subcritical regime where the oil blocks do not percolate, there are three critical curves in the cone separating two localized phases and two delocalized phases, and meeting at two tricritical points. The different phases are characterized by different behavior of the copolymer inside the four neighboring pairs of blocks.
The present paper provides an overview of results obtained in four recent papers by the authors. These papers address the problem of intermittency for the Parabolic Anderson Model in a \emph{time-dependent random medium}, describing the evolution of a ``reactant'' in the presence of a ``catalyst''. Three examples of catalysts are considered: (1) independent simple random walks; (2) symmetric exclusion process; (3) symmetric voter model. The focus is on the annealed Lyapunov exponents, i.e., the exponential growth rates of the successive moments of the reactant. It turns out that these exponents exhibit an interesting dependence on the dimension and on the diffusion constant.