Strong and Weak Disorder for Lévy Directed Polymers in Random Environment

Miura, Mitsuharu; Tawara, Yoshihiro; Tsuchida, Kaneharu

doi:10.1080/07362990802286418

Cited by 6 publications

(4 citation statements)

References 4 publications

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“…Indeed, one might consider the model in which the random walk S is not the simple symmetric random walk on Z d , but belongs to the domain of attraction of an α-stable law with α ∈ (0, 2], see [12]. Let us consider the case of the dimension 1 + 1: it has been showed that weak disorder holds for β small enough when α ∈ (0, 1), see [12], and that strong disorder holds for any β > 0 when α ∈ (1, 2), see [25] (in a continuous setting). A similar question has been studied in [7], where a disordered scaling limit can be constructed whenever α ∈ (1,2].…”

Section: Main Resultmentioning

confidence: 99%

The high-temperature behavior for the directed polymer in dimension $1+2$

Berger¹,

Lacoin²

2017

Ann. Inst. H. Poincaré Probab. Statist.

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Abstract. We investigate the high-temperature behavior of the directed polymer model in dimension 1 + 2. More precisely we study the difference ∆F(β) between the quenched and annealed free energies for small values of the inverse temperature β. This quantity is associated to localization properties of the polymer trajectories, and is related to the overlap fraction of two replicas. Adapting recent techniques developed by the authors in the context of the disordered pinning model [4], we identify the sharp asymptotic high temperature behavior2010 Mathematics Subject Classification: 82D60, 60K37, 82B44.

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Section: Main Resultmentioning

confidence: 99%

The high-temperature behavior for the directed polymer in dimension $1+2$

Berger¹,

Lacoin²

2017

Ann. Inst. H. Poincaré Probab. Statist.

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“…In [44], a continuous version of α-stable longrange polymers is considered. Specifically, a phase transition was shown for the normalized partition function associated to a Lévy process subjected to a Poissonian random environment.…”

Section: Continuous Versions Of Long-range Polymersmentioning

confidence: 99%

Localization of directed polymers with general reference walk

Bates¹

2018

Electron. J. Probab.

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Directed polymers in random environment have usually been constructed with a simple random walk on the integer lattice. It has been observed before that several standard results for this model continue to hold for a more general reference walk. Some finer results are known for the so-called long-range directed polymer in which the reference walk lies in the domain of attraction of an α-stable process. In this note, low-temperature localization properties recently proved for the classical case are shown to be true with any reference walk. First, it is proved that the polymer's endpoint distribution is asymptotically purely atomic, thus strengthening the best known result for long-range directed polymers. A second result proving geometric localization along a positive density subsequence is new to the general case. The proofs use a generalization of the approach introduced by the author with S. Chatterjee in a recent manuscript on the quenched endpoint distribution; this generalization allows one to weaken assumptions on the both the walk and the environment. The methods of this paper also give rise to a variational formula for free energy which is analogous to the one obtained in the simple random walk case.2010 Mathematics Subject Classification. 60K37, 82B26, 82B44, 82D60.

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“…Most of the literature concerning the study of directed polymers associates it with a simple symmetric random walk [8,6,7,15] or when the distribution of the increments belongs to the domain of attraction of an α-stable law for some α ∈ (0, 2] [11,18,25].…”

Section: Introductionmentioning

confidence: 99%

Directed Polymer for very heavy tailed random walks

Viveros¹

2020

Preprint

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In the present work, we investigate the case of Directed Polymer in a Random Environment (DPRE), when the increments of the random walk are heavy-tailed with tail-exponent equal to zero (P[|X1| ≥ n] decays slower than any power of n). This case has not yet been studied in the context of directed polymers and present key differences with the simple symmetric random walk case and the cases where the increments belong to the domain of attraction of an α-stable law, where α ∈ (0, 2]. We establish the absence of a very strong disorder regime -that is, the free energy equals zero at every temperature -for every disorder distribution. We also prove that a strong disorder regime (partition function converging to zero at low temperature) may exist or not depending on finer properties of the random walk: we establish non-matching necessary and sufficient conditions for having a phase transition from weak to strong disorder. In particular our results imply that for this directed polymer model, very strong disorder is not equivalent to strong disorder, shedding a new light on a long standing conjecture concerning the original nearest-neighbor DPRE.

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Strong and Weak Disorder for Lévy Directed Polymers in Random Environment

Abstract: We consider Lévy directed polymers in the Poisson random environment. We give conditions for strong or weak disorder in terms of the Lévy exponent of symmetric Lévy process.

Cited by 6 publications

References 4 publications

The high-temperature behavior for the directed polymer in dimension $1+2$

The high-temperature behavior for the directed polymer in dimension $1+2$

Localization of directed polymers with general reference walk

Directed Polymer for very heavy tailed random walks

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