arXiv subjects

Frank den Hollander

Publications and source records attributed to Frank den Hollander.

107 records · Page 6Linked to original sources

arXiv2012

Pinning of a random walk by a random walk: proof of a conjecture

In [3] the radius of convergence of the generating function of the collision local time of two independent copies of an irreducible, symmetric and transient random walk on Zd, d \geq 1, was studied. Two versions were considered: z1, the radius of convergence when one walk is averaged over and the other is kept fixed; z2, the radius of convergence when both walks are averaged over. While z2 can be easily computed, no explicit expression is available for z1. In [3] it was conjectured that z1 > z2 under a weak regularity assumption on the random walk. However, this gap was only proved for strongly transient random walk. Subsequently, in [1], [4] and [5] the gap was proved for a subclass of random walks that are transient but not strongly transient. In the present note we settle the full conjecture. The proof is based on a comparison of variational formulas for z1 and z2 derived in [3]. Under additional weak regularity assumptions on the random walk, we derive an upper bound for z1. We further show that the gap persists when the random walk has exponential waiting times. The gaps imply the presence of an intermediate phase for a model of two random walks with a pinning interaction. As explained in [3], they also imply the presence of intermediate phases in three classes of interacting stochastic systems: coupled branching processes, interacting diffusions, and directed polymers with bulk disorder.

math.PR↗

arXiv2012

Lectures on random polymers

These lecture notes are a guided tour through the fascinating world of polymer chains interacting with themselves and/or with their environment. The focus is on the mathematical description of a number of physical and chemical phenomena, with particular emphasis on phase transitions and space-time scaling. The topics covered, though only a selection, are typical for the area. Sections 1-3 describe models of polymers without disorder, Sections 4-6 models of polymers with disorder. Appendices A-E contain tutorials in which a number of key techniques are explained in more detail.

math.PR↗

arXiv2012

Free energy of a copolymer in a micro-emulsion

In this paper we consider a two-dimensional model of a copolymer consisting of a random concatenation of hydrophilic and hydrophobic monomers, immersed in a micro-emulsion of random droplets of oil and water. The copolymer interacts with the micro-emulsion through an interaction Hamiltonian that favors matches and disfavors mismatches between the monomers and the solvents, in such a way that the interaction with the oil is stronger than with the water. The configurations of the copolymers are directed self-avoiding paths in which only steps up, down and right are allowed. The configurations of the micro-emulsion are square blocks with oil and water arranged in percolation-type fashion. The only restriction imposed on the path is that in every column of blocks its vertical displacement on the block scale is bounded. The way in which the copolymer enters and exits successive columns of blocks is a directed self-avoiding path as well, but on the block scale. We refer to this path as the coarse-grained self-avoiding path. We are interested in the limit as the copolymer and the blocks become large, in such a way that the copolymer spends a long time in each block yet visits many blocks. This is a coarse-graining limit in which the space-time scales of the copolymer and of the micro-emulsion become separated. We derive a variational formula for the quenched free energy per monomer, where quenched means that the disorder in the copolymer and the disorder in the micro-emulsion are both frozen. In a sequel paper we will analyze this variational formula and identify the phase diagram. It turns out that there are two regimes, supercritical and subcritical, depending on whether the oil blocks percolate or not along the coarse-grained self-avoiding path. The phase diagrams in the two regimes turn out to be completely different.

math.PR↗

arXiv2012

Variational description of Gibbs-non-Gibbs dynamical transitions for the Curie-Weiss model

We perform a detailed study of Gibbs-non-Gibbs transitions for the Curie-Weiss model subject to independent spin-flip dynamics ("infinite-temperature" dynamics). We show that, in this setup, the program outlined in van Enter, Fernández, den Hollander and Redig can be fully completed, namely that Gibbs-non-Gibbs transitions are equivalent to bifurcations in the set of global minima of the large-deviation rate function for the trajectories of the magnetization conditioned on their endpoint. As a consequence, we show that the time-evolved model is non-Gibbs if and only if this set is not a singleton for some value of the final magnetization. A detailed description of the possible scenarios of bifurcation is given, leading to a full characterization of passages from Gibbs to non-Gibbs -and vice versa- with sharp transition times (under the dynamics Gibbsianness can be lost and can be recovered). Our analysis expands the work of Ermolaev and Kulske who considered zero magnetic field and finite-temperature spin-flip dynamics. We consider both zero and non-zero magnetic field but restricted to infinite-temperature spin-flip dynamics. Our results reveal an interesting dependence on the interaction parameters, including the presence of forbidden regions for the optimal trajectories and the possible occurrence of overshoots and undershoots in the optimal trajectories. The numerical plots provided are obtained with the help of MATHEMATICA.

math.PR↗

arXiv2011

A crossover for the bad configurations of random walk in random scenery

In this paper, we consider a random walk and a random color scenery on Z. The increments of the walk and the colors of the scenery are assumed to be i.i.d. and to be independent of each other. We are interested in the random process of colors seen by the walk in the course of time. Bad configurations for this random process are the discontinuity points of the conditional probability distribution for the color seen at time zero given the colors seen at all later times. We focus on the case where the random walk has increments 0, +1 or -1 with probability epsilon, (1-epsilon)p and (1-epsilon)(1-p), respectively, with p in [1/2,1] and epsilon in [0,1), and where the scenery assigns the color black or white to the sites of Z with probability 1/2 each. We show that, remarkably, the set of bad configurations exhibits a crossover: for epsilon=0 and p in (1/2,4/5) all configurations are bad, while for (p,epsilon) in an open neighborhood of (1,0) all configurations are good. In addition, we show that for epsilon=0 and p=1/2 both bad and good configurations exist. We conjecture that for all epsilon in [0,1) the crossover value is unique and equals 4/5. Finally, we suggest an approach to handle the seemingly more difficult case where epsilon>0 and p in [1/2,4/5), which will be pursued in future work.

math.PR↗

arXiv2011

Quenched Lyapunov exponent for the parabolic Anderson model in a dynamic random environment

We continue our study of the parabolic Anderson equation $\partial u/\partial t = κΔu + γξu$ for the space-time field $u\colon\,\Z^d\times [0,\infty)\to\R$, where $κ\in [0,\infty)$ is the diffusion constant, $Δ$ is the discrete Laplacian, $γ\in (0,\infty)$ is the coupling constant, and $ξ\colon\,\Z^d\times [0,\infty)\to\R$ is a space-time random environment that drives the equation. The solution of this equation describes the evolution of a "reactant" $u$ under the influence of a "catalyst" $ξ$, both living on $\Z^d$. In earlier work we considered three choices for $ξ$: independent simple random walks, the symmetric exclusion process, and the symmetric voter model, all in equilibrium at a given density. We analyzed the \emph{annealed} Lyapunov exponents, i.e., the exponential growth rates of the successive moments of $u$ w.r.t.\ $ξ$, and showed that these exponents display an interesting dependence on the diffusion constant $κ$, with qualitatively different behavior in different dimensions $d$. In the present paper we focus on the \emph{quenched} Lyapunov exponent, i.e., the exponential growth rate of $u$ conditional on $ξ$. We first prove existence and derive some qualitative properties of the quenched Lyapunov exponent for a general $ξ$ that is stationary and ergodic w.r.t.\ translations in $\Z^d$ and satisfies certain noisiness conditions. After that we focus on the three particular choices for $ξ$ mentioned above and derive some more detailed properties. We close by formulating a number of open problems.

math.PR↗

arXiv2011

Kawasaki dynamics with two types of particles: stable/metastable configurations and communication heights

This is the second 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 involve three model-dependent quantities: the energy, the shape and the number of critical droplets. In this second paper we prove the first and the second hypothesis and identify the energy of critical droplets. The paper deals with understanding the geometric properties of subcritical, critical and supercritical droplets, which are crucial in determining the metastable behavior of the system. 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↗

arXiv2010

A large-deviation view on dynamical Gibbs-non-Gibbs transitions

We develop a space-time large-deviation point of view on Gibbs-non-Gibbs transitions in spin systems subject to a stochastic spin-flip dynamics. Using the general theory for large deviations of functionals of Markov processes outlined in Feng and Kurtz [11], we show that the trajectory under the spin-flip dynamics of the empirical measure of the spins in a large block in Z^d satisfies a large deviation principle in the limit as the block size tends to infinity. The associated rate function can be computed as the action functional of a Lagrangian that is the Legendre transform of a certain non-linear generator, playing a role analogous to the moment-generating function in the Gartner-Ellis theorem of large deviation theory when this is applied to finite-dimensional Markov processes. This rate function is used to define the notion of "bad empirical measures", which are the discontinuity points of the optimal trajectories (i.e., the trajectories minimizing the rate function) given the empirical measure at the end of the trajectory. The dynamical Gibbs-non-Gibbs transitions are linked to the occurrence of bad empirical measures: for short times no bad empirical measures occur, while for intermediate and large times bad empirical mea- sures are possible. A future research program is proposed to classify the various possible scenarios behind this crossover, which we refer to as a "nature-versus-nurture" transition.

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arXiv2008

On the localized phase of a copolymer in an emulsion: subcritical percolation regime

The present paper is a continuation of \cite{dHP07b}. The object of interest is a two-dimensional model of a directed copolymer, consisting of a random concatenation of hydrophobic and hydrophilic monomers, immersed in an emulsion, consisting of large blocks of oil and water arranged in a percolation-type fashion. The copolymer interacts with the emulsion through an interaction Hamiltonian that favors matches and disfavors mismatches between the monomers and the solvents, in such a way that the interaction with the oil is stronger than with the water. The model has two regimes, supercritical and subcritical, depending on whether the oil blocks percolate or not. In \cite{dHP07b} we focussed on the supercritical regime and obtained a complete description of the phase diagram, which consists of two phases separated by a single critical curve. In the present paper we focus on the subcritical regime and show that the phase diagram consists of four phases separated by three critical curves meeting in two tricritical points.

math.PR↗

arXiv2008

Homogeneous nucleation for Glauber and Kawasaki dynamics in large volumes at low temperatures

In this paper we study metastability in large volumes at low temperatures. We consider both Ising spins subject to Glauber spin-flip dynamics and lattice gas particles subject to Kawasaki hopping dynamics. Let $\b$ denote the inverse temperature and let $Ł_\b \subset \Z^2$ be a square box with periodic boundary conditions such that $\lim_{\b\to\infty}|Ł_\b|=\infty$. We run the dynamics on $Ł_\b$ starting from a random initial configuration where all the droplets (= clusters of plus-spins, respectively, clusters of particles)are small. For large $\b$, and for interaction parameters that correspond to the metastable regime, we investigate how the transition from the metastable state (with only small droplets) to the stable state (with one or more large droplets) takes place under the dynamics. This transition is triggered by the appearance of a single \emph{critical droplet} somewhere in $Ł_\b$. Using potential-theoretic methods, we compute the \emph{average nucleation time} (= the first time a critical droplet appears and starts growing) up to a multiplicative factor that tends to one as $\b\to\infty$. It turns out that this time grows as $Ke^{Γ\b}/|Ł_\b|$ for Glauber dynamics and $K\b e^{Γ\b}/|Ł_\b|$ for Kawasaki dynamics, where $Γ$ is the local canonical, respectively, grand-canonical energy to create a critical droplet and $K$ is a constant reflecting the geometry of the critical droplet, provided these times tend to infinity (which puts a growth restriction on $|Ł_\b|$). The fact that the average nucleation time is inversely proportional to $|Ł_\b|$ is referred to as \emph{homogeneous nucleation}, because it says that the critical droplet for the transition appears essentially independently in small boxes that partition $Ł_\b$.

math.PR↗

arXiv2008

How T-cells use large deviations to recognize foreign antigens

A stochastic model for the activation of T-cells is analysed. T-cells are part of the immune system and recognize foreign antigens against a background of the body's own molecules. The model under consideration is a slight generalization of a model introduced by Van den Berg, Rand and Burroughs in 2001, and is capable of explaining how this recognition works on the basis of rare stochastic events. With the help of a refined large deviation theorem and numerical evaluation it is shown that, for a wide range of parameters, T-cells can distinguish reliably between foreign antigens and self-antigens.

q-bio.SC↗

arXiv2008

Invasion percolation on regular trees

We consider invasion percolation on a rooted regular tree. For the infinite cluster invaded from the root, we identify the scaling behavior of its $r$-point function for any $r\geq2$ and of its volume both at a given height and below a given height. We find that while the power laws of the scaling are the same as for the incipient infinite cluster for ordinary percolation, the scaling functions differ. Thus, somewhat surprisingly, the two clusters behave differently; in fact, we prove that their laws are mutually singular. In addition, we derive scaling estimates for simple random walk on the cluster starting from the root. We show that the invasion percolation cluster is stochastically dominated by the incipient infinite cluster. Far above the root, the two clusters have the same law locally, but not globally. A key ingredient in the proofs is an analysis of the forward maximal weights along the backbone of the invasion percolation cluster. These weights decay toward the critical value for ordinary percolation, but only slowly, and this slow decay causes the scaling behavior to differ from that of the incipient infinite cluster.

math.PR↗

arXiv2007

On the localized phase of a copolymer in an emulsion: supercritical percolation regime

In this paper we study a two-dimensional directed self-avoiding walk model of a random copolymer in a random emulsion. The copolymer is a random concatenation of monomers of two types, $A$ and $B$, each occurring with density 1/2. The emulsion is a random mixture of liquids of two types, $A$ and $B$, organised in large square blocks occurring with density $p$ and $1-p$, respectively, where $p \in (0,1)$. The copolymer in the emulsion has an energy that is minus $α$ times the number of $AA$-matches minus $β$ times the number of $BB$-matches, where without loss of generality the interaction parameters can be taken from the cone $\{(α,β)\in\R^2\colon α\geq |β|\}$. To make the model mathematically tractable, we assume that the copolymer is directed and can only enter and exit a pair of neighbouring blocks at diagonally opposite corners. In \cite{dHW06}, it was found that in the supercritical percolation regime $p \geq p_c$, with $p_c$ the critical probability for directed bond percolation on the square lattice, the free energy has a phase transition along a curve in the cone that is independent of $p$. At this critical curve, there is a transition from a phase where the copolymer is fully delocalized into the $A$-blocks to a phase where it is partially localized near the $AB$-interface. In the present paper we prove three theorems that complete the analysis of the phase diagram : (1) the critical curve is strictly increasing; (2) the phase transition is second order; (3) the free energy is infinitely differentiable throughout the partially localized phase.

math.PR↗

arXiv2007

The renormalization transformation for two-type branching models

This paper studies countable systems of linearly and hierarchically interacting diffusions taking values in the positive quadrant. These systems arise in population dynamics for two types of individuals migrating between and interacting within colonies. Their large-scale space-time behavior can be studied by means of a renormalization program. This program, which has been carried out successfully in a number of other cases (mostly one-dimensional), is based on the construction and the analysis of a nonlinear renormalization transformation, acting on the diffusion function for the components of the system and connecting the evolution of successive block averages on successive time scales. We identify a general class of diffusion functions on the positive quadrant for which this renormalization transformation is well-defined and, subject to a conjecture on its boundary behavior, can be iterated. Within certain subclasses, we identify the fixed points for the transformation and investigate their domains of attraction. These domains of attraction constitute the universality classes of the system under space-time scaling.

math.PR↗

arXiv2006

Dynamics & Stochastics: Festschrift in honor of M. S. Keane

The present volume is a Festschrift for Mike Keane, on the occasion of his 65th birthday on January 2, 2005. It contains 29 contributions by Mike's closest colleagues and friends, covering a broad range of topics in Dynamics and Stochastics. To celebrate Mike's scientific achievements, a conference entitled ``Dynamical Systems, Probability Theory and Statistical Mechanics'' was organized in Eindhoven, The Netherlands, during the week of January 3--7, 2005. This conference was hosted jointly by EURANDOM and by Philips Research. It drew over 80 participants from 5 continents, which is a sign of the warm affection and high esteem for Mike felt by the international mathematics community.

math.PR↗

arXiv2006

Random walk in random scenery: A survey of some recent results

. In this paper we give a survey of some recent results for random walk in random scenery (RWRS). On $\mathbb {Z}^d$, $d\geq 1$, we are given a random walk with i.i.d. increments and a random scenery with i.i.d. components. The walk and the scenery are assumed to be independent. RWRS is the random process where time is indexed by $\mathbb {Z}$, and at each unit of time both the step taken by the walk and the scenery value at the site that is visited are registered. We collect various results that classify the ergodic behavior of RWRS in terms of the characteristics of the underlying random walk (and discuss extensions to stationary walk increments and stationary scenery components as well). We describe a number of results for scenery reconstruction and close by listing some open questions.

math.PR↗

arXiv2001

Moderate deviations for the volume of the Wiener sausage

For a>0,let W^a(t) be the a-neighbourhood of standard Brownian motion in R^d starting at 0 and observed until time t.It is well-known that E|W^a(t)|~kappa_a t (t->infty) for d >= 3,with kappa_a the Newtonian capacity of the ball with radius a. We prove that lim_{t->infty} 1/t^{(d-2)/d}log P(|W^a(t)|<=bt) = -I^{kappa_a}(b) in (-infty,0) for all 0 infty.This is markedly different from the optimal strategy for large deviations |W^a(t)|<=f(t) with f(t)=o(t),where W^a(t) is known to fill completely a ball of volume f(t) and nothing outside,so that W^a(t) has no holes and f(t)^{-1/d}W^a(t) is localised in the limit as t->infty.We give a detailed analysis of the rate function I^{kappa_a},in particular,its behaviour near the boundary points of (0,kappa_a).It turns out that I^{kappa_a} has an infinite slope at kappa_a and,remarkably,for d>=5 is nonanalytic at some critical point in (0,kappa_a),above which it follows a pure power law.This crossover is associated with a collapse transition in the optimal strategy.We also derive the analogous moderate deviation result for d=2.In this case E|W^a(t)|~2pi t/log t (t->infty),and we prove that lim_{t->infty} 1/log t log P(|W^a(t)|<=bt/log t) =-I^{2pi}(b)in (-infty,0) for all 0<b<2pi.

math.PR↗