A model of continuous time polymer on the lattice

Márquez-Carreras, David; Rovira, Carles; Tindel, Samy

doi:10.31390/cosa.5.1.07

Cited by 5 publications

(3 citation statements)

References 26 publications

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“…The function y * T , defined in (20), depends on κ, β and on the environment, it is called the favourite path. Our statement improves on the literature on polymers by concerning the path itself, not only the terminal location at time T as in [6] and [10].…”

Section: Introductionmentioning

confidence: 99%

Overlaps and pathwise localization in the Anderson polymer model

Comets¹,

Cranston²

2013

Stochastic Processes and their Applications

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We consider large time behavior of typical paths under the Anderson polymer measure. If P x κ is the measure induced by rate κ, simple, symmetric random walk on Z d started at x, this measure is defined aswhere {W x : x ∈ Z d } is a field of iid standard, one-dimensional Brownian motions, β > 0, κ > 0 and Z κ,β,t (x) the normalizing constant. We establish that the polymer measure gives a macroscopic mass to a small neighborhood of a typical path as T → ∞, for parameter values outside the perturbative regime of the random walk, giving a pathwise approach to polymer localization, in contrast with existing results. The localization becomes complete as β 2 κ → ∞ in the sense that the mass grows to 1. The proof makes use of the overlap between two independent samples drawn under the Gibbs measure µ x κ,β,T , which can be estimated by the integration by parts formula for the Gaussian environment. Conditioning this measure on the number of jumps, we obtain a canonical measure which already shows scaling properties, thermodynamic limits, and decoupling of the parameters.

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Section: Introductionmentioning

confidence: 99%

Overlaps and pathwise localization in the Anderson polymer model

Comets¹,

Cranston²

2013

Stochastic Processes and their Applications

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“…More recently, however, the idea has been applied to the Sherrington-Kirkpatrick model of spin glasses in [CN95], and then re-applied to greater effect by a series of other authors [BKL02,Tin05]. The paper [MCRT11] also uses the same technique in the context of lattice polymer models, which are somewhat similar to ours through the connection between tree polymers and multiplicative cascades. However, the main purpose of these papers is to use the dynamic weights technique to derive growth exponents and fluctuation behavior for partition functions of Gibbs measures as the size of the system grows large, whereas we are more concerned with showing that the infinite volume measure-valued process has the properties listed above.…”

Section: Introductionmentioning

confidence: 92%

Diffusions of multiplicative cascades

Alberts

Rifkind

2014

Stochastic Processes and their Applications

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A multiplicative cascade can be thought of as a randomization of a measure on the boundary of a tree, constructed from an iid collection of random variables attached to the tree vertices. Given an initial measure with certain regularity properties, we construct a continuous time, measure-valued process whose value at each time is a cascade of the initial one. We do this by replacing the random variables on the vertices with independent increment processes satisfying certain moment assumptions. Our process has a Markov property: at any given time it is a cascade of the process at any earlier time by random variables that are independent of the past. It has the further advantage of being a martingale and, under certain extra conditions, it is also continuous. For Gaussian independent increments processes we develop the infinite-dimensional stochastic calculus that describes the evolution of the measure process, and use it to compute the optimal Hölder exponent in the Wasserstein distance on measures. We also discuss applications of this process to models of tree polymers and one-dimensional random geometry.

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“…The Anderson polymer model is the Gibbs measure onD = D([0, ], Z ) defined by , , ( ) = , , ( ) −1 [ exp { ∫ 0 ( ) ( )}](177)for bounded measurable : D → R, where , , () =[exp{ ∫ 0 ( ) ( )}] is the partition function. This model has received a lot of attention in recent years and a partial list of references would include[33,68,[81][82][83][84][85][86][87][88][89][90][91][92][93][94][95].By the Feynman-Kac formula, of the time-dependent parabolic Anderson equation (or stochastic heat equation)…”

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confidence: 99%

Properties of the Parabolic Anderson Model and the Anderson Polymer Model

Cranston

2013

ISRN Probability and Statistics

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In this article we examine some properties of the solutions of the parabolic Anderson model. In particular we discuss intermittency of the field of solutions of this random partial differential equation, when it occurs and what the field looks like when intermittency doesn't hold. We also explore the behavior of a polymer model created by a Gibbs measure based on solutions to the parabolic Anderson equation.

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A model of continuous time polymer on the lattice

Cited by 5 publications

References 26 publications

Overlaps and pathwise localization in the Anderson polymer model

Overlaps and pathwise localization in the Anderson polymer model

Diffusions of multiplicative cascades

Properties of the Parabolic Anderson Model and the Anderson Polymer Model

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