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Overlaps and Pathwise Localization in the Anderson Polymer Model
, 2012
"... We consider large time behavior of typical paths under the Anderson polymermeasure. IfP x κ isthemeasureinducedbyrateκ,simple, symmetricrandom walk on Zd started at x, this measure is defined as dµ x κ,β,T (X) = Zκ,β,T(x) −1 { ∫ T exp β dWX(s)(s) dP 0 x κ(X) where {Wx: x ∈ Zd} is a field of iid st ..."
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We consider large time behavior of typical paths under the Anderson polymermeasure. IfP x κ isthemeasureinducedbyrateκ,simple, symmetricrandom walk on Zd started at x, this measure is defined as dµ x κ,β,T (X) = Zκ,β,T(x) −1 { ∫ T exp β dWX(s)(s) dP 0 x κ(X) where {Wx: x ∈ Zd} is a field of iid standard, onedimensional Brownian motions, β> 0,κ> 0 and Zκ,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 existingresults. Thelocalization becomes complete as β2 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.
A model of continuous time polymer on the lattice
 In preparation
, 2007
"... Abstract. In this article, we try to give a rather complete picture of the behavior of ..."
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Abstract. In this article, we try to give a rather complete picture of the behavior of
LYAPUNOV EXPONENTS FOR STOCHASTIC ANDERSON MODELS WITH NONGAUSSIAN NOISE
, 2008
"... The stochastic Anderson model in discrete or continuous space is defined for a class of nonGaussian spacetime potentials W as solutions u to the multiplicative stochastic heat equation u(t, x) = 1 + � t 0 κ∆u(s, x)ds + � t 0 βW (ds, x)u(s, x) with diffusivity κ and inversetemperature β. The re ..."
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The stochastic Anderson model in discrete or continuous space is defined for a class of nonGaussian spacetime potentials W as solutions u to the multiplicative stochastic heat equation u(t, x) = 1 + � t 0 κ∆u(s, x)ds + � t 0 βW (ds, x)u(s, x) with diffusivity κ and inversetemperature β. The relation with the corresponding polymer model in a random environment is given. The large time exponential behavior of u is studied via its almost sure Lyapunov exponent λ = limt→ ∞ t−1 log u(t, x), which is proved to exist, and is estimated as a function of β and κ for β2κ−1 bounded below: positivity and nontrivial upper bounds are established, generalizing and improving existing results. In discrete space λ is of order β2 / log � β2 /κ � and in continuous space it is between β2 � κ/β2�H/(H+1) and β 2 � κ/β 2 � H/(2H+1).
Localization Transition for Polymers in Poissonian Medium 1
, 2012
"... We study a model of directed polymers in random environment in dimension 1 + d, given by a Brownian motion in a Poissonian potential. We study the effect of the density and the strength of inhomogeneities, respectively the intensity parameter ν of the Poisson field and the temperature inverse β. Our ..."
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We study a model of directed polymers in random environment in dimension 1 + d, given by a Brownian motion in a Poissonian potential. We study the effect of the density and the strength of inhomogeneities, respectively the intensity parameter ν of the Poisson field and the temperature inverse β. Our results are: (i) fine information on the phase diagram, with quantitative estimates on the critical curve; (ii) pathwise localization at low temperature and/or large density; (iii) complete localization in a favourite corridor for large νβ 2 and bounded β.
Localization Transition for Polymers in Poissonian Medium1 Francis COMETS 2
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