One-dimensional polymers in random environments: stretching vs. folding

Quentin Berger, Chien-Hao Huang, Niccolò Torri, Ran Wei

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
JournalElectronic Journal of Probability
Publication statusPublished - 2022
Externally publishedYes


  • heavy-tail distributions
  • random polymer
  • random walk
  • range
  • sub-diffusivity
  • super-diffusivity
  • weak-coupling limit


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