Localization Transition for Polymers in Poissonian Medium

Comets, Francis; Yoshida, Nobuo

doi:10.1007/s00220-013-1744-8

Cited by 24 publications

(32 citation statements)

References 35 publications

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“…where : exp : denotes the Wick exponential. Renormalization put aside, this equation is similar to the definition of the point-to-point partition of our polymer model (15). Alberts, Khanin and Quastel [1] were indeed able to construct a polymer measure, with P2P partition function given by Z β (T, X).…”

Section: The Continuous Casementioning

confidence: 81%

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The intermediate disorder regime for Brownian directed polymers in Poisson environment

Cosco

2019

Indagationes Mathematicae

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We consider the Brownian directed polymer in Poissonian environment in dimension 1+1, under the so-called intermediate disorder regime [2], which is a crossover regime between the strong and weak disorder regions. We show that, under a diffusive scaling involving different parameters of the system, the renormalized point-to-point partition function of the polymer converges in law to the solution of the stochastic heat equation with Gaussian multiplicative noise. The Poissonian environment provides a natural setting and strong tools, such as the Wiener-Itô chaos expansion [38], which, applied to the partition function, is the basic ingredient of the proof.AMS 2010 subject classifications. 60F05, 82D60.

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Section: The Continuous Casementioning

confidence: 81%

“…Whether or not very strong disorder is equivalent to strong disorder is still an open question. Path localization properties of the polymer were studied by Comets and Yoshida in [15]. For all time t ≥ 0, they considered a favorite path Y (t) s , depending only on the Poisson environment, such that for all s ≤ t,…”

Section: The Model and Its Contextmentioning

confidence: 99%

The intermediate disorder regime for Brownian directed polymers in Poisson environment

Cosco

2019

Indagationes Mathematicae

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“…It is for instance believed that the difference between ϕ(β) and the so-called annealed free energy characterizes the localized/delocalized phases. See [3,8,6,11] for rigorous results in this direction.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…It is possible to show by a block argument that the finite volume free energy stays bounded as β → −∞ in general but, to the best of our knowledge, the existence of the limit at β = −∞ is not known. Later in [11], the asymptotics as the density of the Poisson point process tends to ∞ was also studied but only for bounded β, in contrast to Theorem 2 here.…”

Section: Zero Temperature Limits and Open Paths Countingmentioning

confidence: 99%

Limiting Results for the Free Energy of Directed Polymers in Random Environment with Unbounded Jumps

et al. 2015

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Abstract. We study asymptotics of the free energy for the directed polymer in random environment. The polymer is allowed to make unbounded jumps and the environment is given by Bernoulli variables. We first establish the existence and continuity of the free energy including the negative infinity value of the coupling constant β. Our proof of existence at β = −∞ differs from existing ones in that it avoids the direct use of subadditivity. Secondly, we identify the asymptotics of the free energy at β = −∞ in the limit of the success probability of the Bernoulli variables tending to one. It is described by using the so-called time constant of a certain directed first passage percolation. Our proof relies on a certain continuity property of the time constant, which is of independent interest.

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“…As far as I know, the proof of this result was written down for the first time in Comets [6, Chapter 6], although versions for slightly different models appeared earlier in the works of Comets and Cranston [7] and Comets and Yoshida [8]. In the language of statistical physics, this shows that path localization happens in the annealed sense.…”

Section: 2mentioning

confidence: 92%