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Annealed vs quenched critical points for a random walk pinning
"... Abstract We study a random walk pinning model, where conditioned on a simple random walk Y on Z d acting as a random medium, the path measure of a second independent simple random walk X up to time t is Gibbs transformed with Hamiltonian −L t (X, Y ), where L t (X, Y ) is the collision local time b ..."
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Abstract We study a random walk pinning model, where conditioned on a simple random walk Y on Z d acting as a random medium, the path measure of a second independent simple random walk X up to time t is Gibbs transformed with Hamiltonian −L t (X, Y ), where L t (X, Y ) is the collision local time between X and Y up to time t. This model arises naturally in various contexts, including the study of the parabolic Anderson model with moving catalysts, the parabolic Anderson model with Brownian noise, and the directed polymer model. It falls in the same framework as the pinning and copolymer models, and exhibits a localizationdelocalization transition as the inverse temperature β varies. We show that in dimensions d = 1, 2, the annealed and quenched critical values of β are both 0, while in dimensions d ≥ 4, the quenched critical value of β is strictly larger than the annealed critical value (which is positive).
Disorder relevance for the random walk pinning model in dimension 3
"... We study the continuous time version of the random walk pinning model, where conditioned on a continuous time random walk (Ys)s≥0 on Zd with jump rate ρ> 0, which plays the role of disorder, the law up to time t of a second independent random walk (Xs)0≤s≤t with jump rate 1 is Gibbs transformed w ..."
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We study the continuous time version of the random walk pinning model, where conditioned on a continuous time random walk (Ys)s≥0 on Zd with jump rate ρ> 0, which plays the role of disorder, the law up to time t of a second independent random walk (Xs)0≤s≤t with jump rate 1 is Gibbs transformed with weight eβLt(X,Y), where Lt(X,Y) is the collision local time between X and Y up to time t. As the inverse temperature β varies, the model undergoes a localizationdelocalization transition at some critical βc ≥ 0. A natural question is whether or not there is disorder relevance, namely whether or not βc differs from the critical point βannc for the annealed model. In [BS09], it was shown that there is disorder irrelevance in dimensions d = 1 and 2, and disorder relevance in d ≥ 4. For d ≥ 5, disorder relevance was first proved in [BGdH08]. In this paper, we prove that if X and Y have the same jump probability kernel, which is irreducible and symmetric with finite second moments, then there is also disorder relevance in the critical dimension d = 3, and βc − βannc is at least of the order e−C(ζ)/ρ ζ C(ζ)> 0, for any ζ> 2. Our proof employs coarse graining and fractional moment techniques, which have recently been applied by Lacoin [L09] to the directed polymer model in random environment, and by Giacomin, Lacoin and Toninelli [GLT09] to establish disorder relevance for the random pinning model in the critical dimension. Along the way, we also prove a continuous time version of Doney’s local limit theorem [D97] for renewal processes with infinite mean.
Modeling Flocks and Prices: Jumping Particles with an Attractive Interaction
, 2012
"... We introduce and investigate a new model of a finite number of particles jumping forward on the real line. The jump lengths are independent of everything, but the jump rate of each particle depends on the relative position of the particle compared to the center of mass of the system. The rates are h ..."
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We introduce and investigate a new model of a finite number of particles jumping forward on the real line. The jump lengths are independent of everything, but the jump rate of each particle depends on the relative position of the particle compared to the center of mass of the system. The rates are higher for those left behind, and lower for those ahead of the center of mass, providing an attractive interaction keeping the particles together. We prove that in the fluid limit, as the number of particles goes to infinity, the evolution of the system is described by a mean field equation that exhibits traveling wave solutions. A connection to extreme value statistics is also provided.
Quenched LDP for words in a letter sequence
, 2008
"... When we cut an i.i.d. sequence of letters into words according to an independent renewal process, we obtain an i.i.d. sequence of words. In the annealed large deviation principle (LDP) for the empirical process of words, the rate function is the specific relative entropy of the observed law of words ..."
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When we cut an i.i.d. sequence of letters into words according to an independent renewal process, we obtain an i.i.d. sequence of words. In the annealed large deviation principle (LDP) 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 the present paper we consider the quenched LDP, i.e., we condition on a typical letter sequence. We focus on the case where the renewal process has an algebraic tail. The rate function turns 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, with the former being obtained by concatenating the words and randomising the location of the origin. The proportionality constant equals the tail exponent of the renewal process. Earlier work by Birkner considered the case where the renewal process has an exponential tail, in which case the rate function turns out to be the first term on the set where the second term vanishes and to be infinite elsewhere. We apply our LDP to prove that the radius of convergence of the moment generating function of the collision local time of two strongly transient random walks on Zd, d ≥ 1, strictly increases when we condition on one of the random walks, both in discrete time and in continuous time. The presence of these gaps implies the existence of an intermediate phase for the longtime behaviour of a class of coupled branching processes, interacting diffusions, respectively, directed polymers in random environments.
EURANDOM PREPRINT SERIES
"... A key large deviation principle for interacting stochastic systems ..."
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interaction
, 2012
"... Abstract. We introduce and investigate a new model of a finite number of particles jumping forward on the real line. The jump lengths are independent of everything, but the jump rate of each particle depends on the relative position of the particle compared to the center of mass of the system. The r ..."
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Abstract. We introduce and investigate a new model of a finite number of particles jumping forward on the real line. The jump lengths are independent of everything, but the jump rate of each particle depends on the relative position of the particle compared to the center of mass of the system. The rates are higher for those left behind, and lower for those ahead of the center of mass, providing an attractive interaction keeping the particles together. We prove that in the fluid limit, as the number of particles goes to infinity, the evolution of the system is described by a mean field equation that exhibits traveling wave solutions. A connection to extreme value statistics is also provided. Résumé. Nous introduisons et étudions un nouveau modèle comprenant un nombre fini de particules situées sur la droite réelle et pouvant effectuer des sauts vers la droite. Les longueurs des sauts sont indépendantes du reste, mais le taux de saut de chaque particule dépend de la position relative de la particule par rapport au centre de masse du système. Les taux sont plus grands pour celles qui sont en retard, et plus petits pour celles qui sont en avance par rapport au centre de masse; cela crée ainsi une interaction attractive qui favorise la cohésion des particules. Nous montrons qu’à la limite fluide, lorsque le nombre de particules tend vers l’infini, l’évolution du système est décrite par une équation de champ moyen ayant des solutions d’ondes progressives. On présente