arXiv subjects

Frank den Hollander

Publications and source records attributed to Frank den Hollander.

At least 91 records · Page 5Linked to original sources

arXiv2015

Random Walk on Random Walks

In this paper we study a random walk in a one-dimensional dynamic random environment consisting of a collection of independent particles performing simple symmetric random walks in a Poisson equilibrium with density $ρ\in (0,\infty)$. At each step the random walk performs a nearest-neighbour jump, moving to the right with probability $p_{\circ}$ when it is on a vacant site and probability $p_{\bullet}$ when it is on an occupied site. Assuming that $p_\circ \in (0,1)$ and $p_\bullet \neq \tfrac12$, we show that the position of the random walk satisfies a strong law of large numbers, a functional central limit theorem and a large deviation bound, provided $ρ$ is large enough. The proof is based on the construction of a renewal structure together with a multiscale renormalisation argument.

math.PR↗

arXiv2015

Breaking of ensemble equivalence in networks

It is generally believed that, in the thermodynamic limit, the microcanonical description as a function of energy coincides with the canonical description as a function of temperature. However, various examples of systems for which the microcanonical and canonical ensembles are not equivalent have been identified. A complete theory of this intriguing phenomenon is still missing. Here we show that ensemble nonequivalence can manifest itself also in random graphs with topological constraints. We find that, while graphs with a given number of links are ensemble-equivalent, graphs with a given degree sequence are not. This result holds irrespective of whether the energy is nonadditive (as in unipartite graphs) or additive (as in bipartite graphs). In contrast with previous expectations, our results show that: (1) physically, nonequivalence can be induced by an extensive number of local constraints, and not necessarily by long-range interactions or nonadditivity; (2) mathematically, nonquivalence is determined by a different large-deviation behaviour of microcanonical and canonical probabilities for a single microstate, and not necessarily for almost all microstates. The latter criterion, which is entirely local, is not restricted to networks and holds in general.

cond-mat.stat-mech↗

arXiv2015

Parabolic Anderson model in a dynamic random environment: random conductances

The parabolic Anderson model is defined as the partial differential equation \partial u(x,t)/\partial t = κΔu(x,t) + ξ(x,t)u(x,t), x\in\Z^d, t\geq 0, where κ\in [0,\infty) is the diffusion constant, Δis the discrete Laplacian, and ξis a dynamic random environment that drives the equation. The initial condition u(x,0)=u_0(x), x\in\Z^d, is taken to be non-negative and bounded. The solution of the parabolic Anderson equation describes the evolution of a field of particles performing independent simple random walks with binary branching: particles jump at rate 2dκ, split into two at rate ξ\vee 0, and die at rate (-ξ) \vee 0. Our focus is on the Lyapunov exponents λ_p(κ) = \lim_{t\to\infty} \frac{1}{t} \log \E([u(0,t)]^p)^{1/p}, p \in \N, and λ_0(κ) = \lim_{t\to\infty} \frac{1}{t}\log u(0,t). We investigate what happens when κΔis replaced by Δ^\cK, where \cK = \{\mathcal{K}(x,y)\colon\,x,y\in\Z^d,\,x \sim y\} is a collection of random conductances between neighbouring sites replacing the constant conductances κin the homogeneous model. We show that the associated annealed Lyapunov exponents are given by the formula λ_p(\cK) = \sup\{λ_p(κ) \colon\,κ\in\Supp(\cK)\}, where \Supp(\cK) is the set of values taken by the \cK-field. We also show that for the associated quenched Lyapunov exponent this formula only provides a lower bound. Our proof is valid for three classes of reversible ξ, and for all \cK satisfying a certain clustering property, namely, there are arbitrarily large balls where \cK is almost constant and close to any value in \Supp(\cK). What our result says is that the Lyapunov exponents are controlled by those pockets of \cK where the conductances are close to the value that maximises the growth in the homogeneous setting.

math.PR↗

arXiv2014

Gibbs-non-Gibbs dynamical transitions for mean-field interacting Brownian motions

We consider a system of real-valued spins interacting with each other through a mean-field Hamiltonian that depends on the empirical magnetization of the spins via a general potential. The system is subjected to a stochastic dynamics where the spins perform independent Brownian motions. As in \cite{FedHoMa13}, which considers the Curie-Weiss model with Ising spins interacting via a quadratic potential and subjected to independent spins flips, we follow the program outlined in \cite{vEFedHoRe10}. We show that in the thermodynamic limit the system is non-Gibbs at time $t \in (0,\infty)$ if and only if there exists an $α\in \mathbb{R}$ such that the large deviation rate function for the trajectory of the magnetization conditional on hitting the value $α$ at time $t$ has multiple global minimizers. We further show that different minimizing trajectories are different at time $t=0$. We give conditions on the potential under which the system is Gibbs at time $t=0$, classify the possible scenarios of being Gibbs at time $t \in (0,\infty)$ in terms of the second difference quotient of the potential, and show that the system cannot become Gibbs once it has become non-Gibbs, i.e., there is a unique and explicitly computable crossover time $t_c \in [0,\infty]$ from Gibbs to non-Gibbs. We give examples of immediate loss of Gibbsianness ($t_c=0$), short-time conservation of Gibbsianness, large-time loss of Gibbsianness ($t_c\in (0,\infty)$), and preservation of Gibbsianness ($t_c=\infty$). Depending on the potential, the system can be Gibbs or non-Gibbs at the cross-over time time $t=t_c$.

math.PR↗

arXiv2014

Renormalisation of hierarchically interacting Cannings processes

The present paper brings a new class of interacting jump processes into focus. We start from a single-colony $C^Λ$-process, which arises as the continuum-mass limit of a $Λ$-Cannings individual-based population model, where $Λ$ is a finite non-negative measure that describes the offspring mechanism. After that we introduce a system of hierarchically interacting $C^Λ$-processes, where the interaction comes from migration and reshuffling-resampling based on measures $(Λ_k)_{k}$ both acting in $k$-blocks of the hierarchical group. We refer to this system as the $C_N^{c,Λ}$-process. The dual process of the $C_N^{c,Λ}$-process is a spatial coalescent with multi-level block coalescence. For the above system we carry out a full renormalisation analysis in the hierarchical mean-field limit $N\to\infty$. Our main result is that, in the limit as $N\to\infty$, on each scale $k\in\mathbb{N}_0$ the $k$-block averages of the $C_N^{c,Λ}$-process converge to a random process that is a superposition of a $C^{Λ_k}$-process and a Fleming-Viot process, the latter with a volatility $d_k$ and with a drift of strength $c_k$ towards the limiting $(k+1)$-block average. It turns out that $d_k$ is a function of $c_l$ and $Λ_l$ for all $0\leq l<k$. Thus, it is through the volatility that the renormalisation manifests itself. We discuss the implications of the scaling of $d_k$ for the behaviour on large space-time scales of the $C_N^{c,Λ}$-process. We compare the outcome with what is known from the renormalisation analysis of hierarchically interacting Fleming-Viot diffusions, pointing out several new features. We obtain a new classification for when the process exhibits clustering, respectively, exhibits local coexistence. Finally, we show that for finite $N$ the same dichotomy between clustering and local coexistence holds as for $N\to\infty$.

math.PR↗

arXiv2013

A quenched large deviation principle in a continuous scenario

We prove the analogue for continuous space-time of the quenched LDP derived in Birkner, Greven and den Hollander (2010) for discrete space-time. In particular, we consider a random environment given by Brownian increments, cut into pieces according to an independent continuous-time renewal process. We look at the empirical process obtained by recording both the length of and the increments in the successive pieces. For the case where the renewal time distribution has a Lebesgue density with a polynomial tail, we derive the quenched LDP for the empirical process, i.e., the LDP conditional on a typical environment. The rate function is a sum of two specific relative entropies, one for the pieces and one for the concatenation of the pieces. We also obtain a quenched LDP when the tail decays faster than algebraic. The proof uses coarse-graining and truncation arguments, involving various approximations of specific relative entropies that are not quite standard. In a companion paper we show how the quenched LDP and the techniques developed in the present paper can be applied to obtain a variational characterisation of the free energy and the phase transition line for the Brownian copolymer near a selective interface.

math.PR↗

arXiv2013

Large deviation principles for words drawn from correlated letter sequences

When an i.i.d.\ sequence of letters is cut into words according to i.i.d.\ renewal times, an i.i.d.\ sequence of words is obtained. In the \emph{annealed} LDP (large deviation principle) for the empirical process of words, the rate function is the specific relative entropy of the observed law of words w.r.t.\ the reference law of words. In Birkner, Greven and den Hollander \cite{BGdH10} the \emph{quenched} LDP (= conditional on a typical letter sequence) was derived for the case where the renewal times have an \emph{algebraic} tail. The rate function turned out to be a sum of two terms, one being the annealed rate function, the other being proportional to the specific relative entropy of the observed law of letters w.r.t.\ the reference law of letters, obtained by concatenating the words and randomising the location of the origin. The proportionality constant equals the tail exponent of the renewal process. The purpose of the present paper is to extend both LDP's to letter sequences that are not i.i.d. It is shown that both LDP's carry over when the letter sequence satisfies a mixing condition called \emph{summable variation}. The rate functions are again given by specific relative entropies w.r.t.\ the reference law of words, respectively, letters. But since neither of these reference laws is i.i.d., several approximation arguments are needed to obtain the extension.

math.PR↗

arXiv2013

Variational description of Gibbs-non-Gibbs dynamical transitions for spin-flip systems with a Kac-type interaction

We continue our study of Gibbs-non-Gibbs dynamical transitions. In the present paper we consider a system of Ising spins on a large discrete torus with a Kac-type interaction subject to an independent spin-flip dynamics (infinite-temperature Glauber dynamics). We show that, in accordance with the program outlined in \cite{vEFedHoRe10}, in the thermodynamic limit Gibbs-non-Gibbs dynamical transitions are \emph{equivalent} to bifurcations in the set of global minima of the large-deviation rate function for the trajectories of the empirical density \emph{conditional} on their endpoint. More precisely, the time-evolved measure is non-Gibbs if and only if this set is not a singleton for \emph{some} value of the endpoint. A partial description of the possible scenarios of bifurcation is given, leading to a characterization of passages from Gibbs to non-Gibbs and vice versa, with sharp transition times. Our analysis provides a conceptual step-up from our earlier work on Gibbs-non-Gibbs dynamical transitions for the Curie-Weiss model, where the mean-field interaction allowed us to focus on trajectories of the empirical magnetization rather than the empirical density.

math-ph↗

arXiv2013

Extremal geometry of a Brownian porous medium

The path W[0,t] of a Brownian motion on a d-dimensional torus T^d run for time t is a random compact subset of T^d. We study the geometric properties of the complement T^d \ W[0,t] for t large and d >= 3. In particular, we show that the largest regions in this complement have a linear scale phi = [(d log t)/(d-2)kt]^{1/(d-2)}, where k is the capacity of the unit ball. More specifically, we identify the sets E for which T^d \ W[0,t] contains a translate of phi E, and we count the number of disjoint such translates. Furthermore, we derive large deviation principles for the largest inradius of T^d \ W[0,t] for t large and the epsilon-cover time of T^d for epsilon small. Our results, which generalise laws of large numbers proved by Dembo, Peres and Rosen, are based on a large deviation principle for the shape of the component with largest capacity in T^d \ W_rho[0,t], where W_rho[0,t] is the Wiener sausage of radius rho = rho(t), with rho(t) chosen much smaller than phi but not too small. The idea behind this choice is that T^d \ W[0,t] consists of "lakes", whose linear size is of order phi, connected by narrow "channels". We also derive large deviation principles for the principal Dirichlet eigenvalue and for the maximal volume of the components of T^d \ W_rho[0,t] for t large. Our results give a complete picture of the extremal geometry of T^d \ W[0,t] and of the optimal strategy for W[0,t] to realise the extremes.

math.PR↗

arXiv2013

The parabolic Anderson model in a dynamic random environment: space-time ergodicity for the quenched Lyapunov exponent

We continue our study of the parabolic Anderson equation $\partial u(x,t)/\partial t = κΔu(x,t) + ξ(x,t)u(x,t)$, $x\in\Z^d$, $t\geq 0$, where $κ\in [0,\infty)$ is the diffusion constant, $Δ$ is the discrete Laplacian, and $ξ$ plays the role of a \emph{dynamic random environment} that drives the equation. The initial condition $u(x,0)=u_0(x)$, $x\in\Z^d$, is taken to be non-negative and bounded. The solution of the parabolic Anderson equation describes the evolution of a field of particles performing independent simple random walks with binary branching: particles jump at rate $2dκ$, split into two at rate $ξ\vee 0$, and die at rate $(-ξ) \vee 0$. We assume that $ξ$ is stationary and ergodic under translations in space and time, is not constant and satisfies $\E(|ξ(0,0)|)<\infty$, where $\E$ denotes expectation w.r.t.\ $ξ$. Our main object of interest is the quenched Lyapunov exponent $λ_0 (κ) = \lim_{t\to\infty} \frac{1}{t}\log u(0,t)$. In earlier work we showed that under certain mild space-time mixing assumptions the limit exists $ξ$-a.s., is finite and continuous on $[0,\infty)$, is globally Lipschitz on $(0,\infty)$, is not Lipschitz at 0, and satisfies $λ_0(0) = \E(ξ(0,0))$ and $λ_0(κ) > \E(ξ(0,0))$ for $κ\in (0,\infty)$.In the present paper we show that $\lim_{κ\to\infty} λ_0(κ) =\E(ξ(0,0))$ under an additional space-time mixing condition on $ξ$. This result shows that the parabolic Anderson model exhibits space-time ergodicity in the limit of large diffusivity. This fact is interesting because there are choices of $ξ$ that fulfill our assumption for which the annealed Lyapunov exponent $λ_1(κ) = \lim_{t\to\infty} \frac{1}{t}\log \E(u(0,t))$ is infinite on $[0,\infty)$, a situation that is referred to as strongly catalytic behavior.

math.PR↗

arXiv2013

A general smoothing inequality for disordered polymers

This note sharpens the smoothing inequality of Giacomin and Toninelli for disordered polymers. This inequality is shown to be valid for any disorder distribution with locally finite exponential moments, and to provide an asymptotically sharp constant for weak disorder. A key tool in the proof is an estimate that compares the effect on the free energy of tilting, respectively, shifting the disorder distribution. This estimate holds in large generality (way beyond disordered polymers) and is of independent interest.

math.PR↗

arXiv2013

Variational characterization of the critical curve for pinning of random polymers

In this paper we look at the pinning of a directed polymer by a one-dimensional linear interface carrying random charges. There are two phases, localized and delocalized, depending on the inverse temperature and on the disorder bias. Using quenched and annealed large deviation principles for the empirical process of words drawn from a random letter sequence according to a random renewal process [Birkner, Greven and den Hollander, Probab. Theory Related Fields 148 (2010) 403-456], we derive variational formulas for the quenched, respectively, annealed critical curve separating the two phases. These variational formulas are used to obtain a necessary and sufficient criterion, stated in terms of relative entropies, for the two critical curves to be different at a given inverse temperature, a property referred to as relevance of the disorder. This criterion in turn is used to show that the regimes of relevant and irrelevant disorder are separated by a unique inverse critical temperature. Subsequently, upper and lower bounds are derived for the inverse critical temperature, from which sufficient conditions under which it is strictly positive, respectively, finite are obtained. The former condition is believed to be necessary as well, a problem that we will address in a forthcoming paper. Random pinning has been studied extensively in the literature. The present paper opens up a window with a variational view. Our variational formulas for the quenched and the annealed critical curve are new and provide valuable insight into the nature of the phase transition. Our results on the inverse critical temperature drawn from these variational formulas are not new, but they offer an alternative approach, that is, flexible enough to be extended to other models of random polymers with disorder.

math.PR↗

arXiv2013

Random walk in a high density dynamic random environment

The goal of this note is to prove a law of large numbers for the empirical speed of a green particle that performs a random walk on top of a field of red particles which themselves perform independent simple random walks on $\Z^d$, $d \geq 1$. The red particles jump at rate 1 and are in a Poisson equilibrium with density $μ$. The green particle also jumps at rate 1, but uses different transition kernels $p'$ and $p''$ depending on whether it sees a red particle or not. It is shown that, in the limit as $μ\to\infty$, the speed of the green particle tends to the average jump under $p'$. This result is far from surprising, but it is non-trivial to prove. The proof that is given in this note is based on techniques that were developed in \cite{KeSi} to deal with spread-of-infection models. The main difficulty is that, due to particle conservation, space-time correlations in the field of red particles decay slowly. This places the problem in a class of random walks in dynamic random environments for which scaling laws are hard to obtain.

math.PR↗

arXiv2013

Scaling of a random walk on a supercritical contact process

A proof is provided of a strong law of large numbers for a one-dimensional random walk in a dynamic random environment given by a supercritical contact process in equilibrium. The proof is based on a coupling argument that traces the space-time cones containing the infection clusters generated by single infections and uses that the random walk eventually gets trapped inside the union of these cones. For the case where the local drifts of the random walk are smaller than the speed at which infection clusters grow, the random walk eventually gets trapped inside a single cone. This in turn leads to the existence of regeneration times at which the random walk forgets its past. The latter are used to prove a functional central limit theorem and a large deviation principle. The qualitative dependence of the speed, the volatility and the rate function on the infection parameter is investigated, and some open problems are mentioned.

math.PR↗

arXiv2013

The parabolic Anderson model in a dynamic random environment: basic properties of the quenched Lyapunov exponent

In this paper we study the parabolic Anderson equation \partial u(x,t)/\partial t=κΔu(x,t)+ξ(x,t)u(x,t), x\in\Z^d, t\geq 0, where the u-field and the ξ-field are \R-valued, κ\in [0,\infty) is the diffusion constant, and $Δ$ is the discrete Laplacian. The initial condition u(x,0)=u_0(x), x\in\Z^d, is taken to be non-negative and bounded. The solution of the parabolic Anderson equation describes the evolution of a field of particles performing independent simple random walks with binary branching: particles jump at rate 2dκ, split into two at rate ξ\vee 0, and die at rate (-ξ)\vee 0. Our goal is to prove a number of basic properties of the solution u under assumptions on $ξ$ that are as weak as possible. Throughout the paper we assume that $ξ$ is stationary and ergodic under translations in space and time, is not constant and satisfies \E(|ξ(0,0)|)<\infty, where \E denotes expectation w.r.t. ξ. Under a mild assumption on the tails of the distribution of ξ, we show that the solution to the parabolic Anderson equation exists and is unique for all κ\in [0,\infty). Our main object of interest is the quenched Lyapunov exponent λ_0(κ)=\lim_{t\to\infty}\frac{1}{t}\log u(0,t). Under certain weak space-time mixing conditions on ξ, we show the following properties: (1)λ_0(κ) does not depend on the initial condition u_0; (2)λ_0(κ)<\infty for all κ\in [0,\infty); (3)κ\mapsto λ_0(κ) is continuous on [0,\infty) but not Lipschitz at 0. We further conjecture: (4)\lim_{κ\to\infty}[λ_p(κ)-λ_0(κ)]=0 for all p\in\N, where λ_p (κ)=\lim_{t\to\infty}\frac{1}{pt}\log\E([u(0,t)]^p) is the p-th annealed Lyapunov exponent. Finally, we prove that our weak space-time mixing conditions on ξare satisfied for several classes of interacting particle systems.

math.PR↗

arXiv2012

Copolymer with pinning: variational characterization of the phase diagram

This paper studies a polymer chain in the vicinity of a linear interface separating two immiscible solvents. The polymer consists of random monomer types, while the interface carries random charges. Both the monomer types and the charges are given by i.i.d. sequences of random variables. The configurations of the polymer are directed paths that can make i.i.d. excursions of finite length above and below the interface. The Hamiltonian has two parts: a monomer-solvent interaction ("copolymer") and a monomer-interface interaction ("pinning"). The quenched and the annealed version of the model each undergo a transition from a localized phase (where the polymer stays close to the interface) to a delocalized phase (where the polymer wanders away from the interface). We exploit the approach developed in [5] and [3] to derive variational formulas for the quenched and the annealed free energy per monomer. These variational formulas are analyzed to obtain detailed information on the critical curves separating the two phases and on the typical behavior of the polymer in each of the two phases. Our main results settle a number of open questions.

math.PR↗

arXiv2012

Kawasaki dynamics with two types of particles: critical droplets

This is the third 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 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 are interested in the transition time to the full box. This transition is triggered by a critical droplet appearing somewhere in the box. In the first paper we identified the parameter range for which the system is metastable, showed that the first entrance distribution on the set of critical droplets is uniform, computed the expected transition time up to and including a multiplicative factor of order one, and proved that the nucleation time divided by its expectation is exponentially distributed, all in the limit of low temperature. These results were proved under three hypotheses, and involved three model-dependent quantities: the energy, the shape and the number of critical droplets. Here prove the third hypothesis and identify the shape and the number of critical droplets. The geometric properties of subcritical, critical and supercritical droplets, which are crucial in determining the metastable behavior of the system are identified. The geometry turns out to be considerably more complex than for Kawasaki dynamics with one type of particle, for which an extensive literature exists. The main motivation behind our work is to understand metastability of multi-type particle systems.

math.PR↗

arXiv2012

Law of large numbers for non-elliptic random walks in dynamic random environments

We prove a law of large numbers for a class of $\Z^d$-valued random walks in dynamic random environments, including non-elliptic examples. We assume for the random environment a mixing property called \emph{conditional cone-mixing} and that the random walk tends to stay inside wide enough space-time cones. The proof is based on a generalization of a regeneration scheme developed by Comets and Zeitouni for static random environments and adapted by Avena, den Hollander and Redig to dynamic random environments. A number of one-dimensional examples are given. In some cases, the sign of the speed can be determined.

math.PR↗