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

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 βω_x ´ h sitting on the range of the random walk, where pω_xq_xPZ 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 ω_x has a finite second moment we find a range size fluctuation exponent ξ “ 2/3.