TY - JOUR

T1 - One-dimensional polymers in random environments

T2 - stretching vs. folding

AU - Berger, Quentin

AU - Huang, Chien-Hao

AU - Torri, Niccolò

AU - Wei, Ran

N1 - Funding Information:
*Q. Berger, N. Torri and R. Wei were supported by a public grant overseen by the French National Research Agency, ANR SWiWS (ANR-17-CE40-0032-02). N. Torri was also supported by the project Labex MME-DII (ANR11-LBX-0023-01). C.-H. Huang was supported by the Ministry of Science and Technology grant MOST 110-2115-M-004-001. †Sorbonne Université, LPSM, Campus Pierre et Marie Curie, case 158, 4 place Jussieu, 75252 Paris Cedex 5, France. E-mail: quentin.berger@sorbonne-universite.fr ‡Department of Mathematical Sciences, National Chengchi University, Taipei 16302, Taiwan. E-mail: haohuang@nccu.edu.tw §Université Paris-Nanterre, 200 Av. de la République, 92001, Nanterre, France and FP2M, CNRS FR 2036. E-mail: niccolo.torri@parisnanterre.fr ¶Department of Mathematics, Nanjing University, 22 Hankou Road, 210093 Nanjing, China. E-mail: weiran@nju.edu.cn
Publisher Copyright:
© 2022, Institute of Mathematical Statistics. All rights reserved.

PY - 2022

Y1 - 2022

N2 - In this article we study a non-directed polymer model on Z, that is a one-dimensional simple random walk placed in a random environment. More precisely, the law of the random walk is modified by the exponential of the sum of potentials βωx ´ h sitting on the range of the random walk, where pωxqxPZ are i.i.d. random variables (the disorder) and β ě 0 (disorder strength) and h P R (external field) are two parameters. When β “ 0, h ą 0, this corresponds to a random walk penalized by its range; when β ą 0, h “ 0, this corresponds to the “standard” polymer model in random environment, except that it is non-directed. In this work, we allow the parameters β, h to vary according to the length of the random walk and we study in detail the competition between the stretching effect of the disorder, the folding effect of the external field (if h ě 0) and the entropy cost of atypical trajectories. We prove a complete description of the (rich) phase diagram and we identify scaling limits of the model in the different phases. In particular, in the case β ą 0, h “ 0 of the non-directed polymer, if ωx has a finite second moment we find a range size fluctuation exponent ξ “ 2/3.

AB - In this article we study a non-directed polymer model on Z, that is a one-dimensional simple random walk placed in a random environment. More precisely, the law of the random walk is modified by the exponential of the sum of potentials βωx ´ h sitting on the range of the random walk, where pωxqxPZ are i.i.d. random variables (the disorder) and β ě 0 (disorder strength) and h P R (external field) are two parameters. When β “ 0, h ą 0, this corresponds to a random walk penalized by its range; when β ą 0, h “ 0, this corresponds to the “standard” polymer model in random environment, except that it is non-directed. In this work, we allow the parameters β, h to vary according to the length of the random walk and we study in detail the competition between the stretching effect of the disorder, the folding effect of the external field (if h ě 0) and the entropy cost of atypical trajectories. We prove a complete description of the (rich) phase diagram and we identify scaling limits of the model in the different phases. In particular, in the case β ą 0, h “ 0 of the non-directed polymer, if ωx has a finite second moment we find a range size fluctuation exponent ξ “ 2/3.

KW - heavy-tail distributions

KW - random polymer

KW - random walk

KW - range

KW - sub-diffusivity

KW - super-diffusivity

KW - weak-coupling limit

UR - http://www.scopus.com/inward/record.url?scp=85144425204&partnerID=8YFLogxK

U2 - 10.1214/22-EJP862

DO - 10.1214/22-EJP862

M3 - Article

AN - SCOPUS:85144425204

SN - 1083-6489

VL - 27

JO - Electronic Journal of Probability

JF - Electronic Journal of Probability

ER -