Localization of a one-dimensional polymer in a repulsive i.i.d. environment - Archive ouverte HAL
Pré-Publication, Document De Travail Année : 2023

Localization of a one-dimensional polymer in a repulsive i.i.d. environment

Localisation d'un polymère uni-dimensionnel dans un environnement i.i.d. répulsif

Résumé

The purpose of this paper is to study a one-dimensional polymer penalized by its range and placed in a random environment ω. The law of the simple symmetric random walk up to time n is modified by the exponential of the sum of βω_z − h sitting on its range, with h and β positive parameters. It is known that, at first order, the polymer folds itself to a segment of optimal size c_h n^(1/3) with c_h = π^(2/3) h^(−1/3). Here we study how disorder influences finer quantities. If the random variables ω_z are i.i.d. with a finite second moment, we prove that the left-most point of the range is located near −u_* n^(1/3) , where u_* ∈ [0, c_h ] is a constant that only depends on the disorder. This contrasts with the homogeneous model (i.e. when β = 0), where the left-most point has a random location between −c_h n^(1/3) and 0. With an additional moment assumption, we are able to show that the left-most point of the range is at distance Un^(2/9) from −u_* n^(1/3) and the right-most point at distance Vn^(2/9) from (c_h − u_*)n^(1/3). Here again, U and V are constants that depend only on ω.
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Dates et versions

hal-04157285 , version 1 (10-07-2023)

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Nicolas Bouchot. Localization of a one-dimensional polymer in a repulsive i.i.d. environment. 2023. ⟨hal-04157285⟩
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