Strong localization and macroscopic atoms for directed polymers

Abstract

Abstract. In this article, we derive strong localization results for directed polymers in random environment. We show that at "low temperature" the polymer measure is asymptotically concentrated at a few points of macroscopic mass (we call these points ǫ-atoms). These results are derived assuming weak conditions on the tail decay of the random environment. MSC: 60K37;82B44

Localization Transition for Polymers in Poissonian Medium

“…The trace {x ∈ R d : µ t (χ s,x ) ≥ 1 − δ} at time t of the (δ, t)-predominant set is reminiscent of the ǫ-atoms discovered in the discrete setting in [39], with ǫ = 1 − δ.…”

New Bounds for the Free Energy of Directed Polymers in Dimension 1 + 1 and 1 + 2

“…The model can be defined and it is of interest also with environments with heavier tails (see e.g. [26]) but we will not consider these cases here.…”

“…We study a directed polymer model introduced by Huse and Henley (in dimension 1 1) [18] with the purpose of investigating impurity-induced domain-wall roughening in the 2D-Ising model. The first mathematical study of directed polymers in random environment was made by Imbrie and Spencer [19], and was followed by numerous authors [19,3,1,23,4,6,9,5,8,26] (for a review on the subject see [7]). Directed polymers in random environment model, in particular, polymer chains in a solution with impurities.…”

Semidirected random polymers: Strong disorder and localization

“…This result extends the picture valid in directed polymers [4] and should be contrasted with the one in [3], where the diffusive behavior was established in the regime of parameters where the annealed and the quenched Lyapounov norms coincide. In [5] we also identify situations where the inequality of the norms holds.…”