Random potentials for pinning models with \nabla and \Delta interactions

Abstract

We consider two models for biopolymers, the $\nabla$ interaction and the $\Delta$ one, both with the Gaussian potential in the random environment. A random field $\varphi:{0,1,...,N}\rightarrow \Bbb{R}^d$ represents the position of the polymer path. The law of the field is given by $\exp(-\sum_i\frac{|\nabla\varphi_i|^2}{2})$ where $\nabla$ is the discrete gradient, and by $\exp(-\sum_i\frac{|\Delta\varphi_i|^2}{2})$ where $\Delta$ is the discrete Laplacian. For every Gaussian potential $\frac{|\cdot|^2}{2}$, a random charge is added as a factor: $(1+\beta\omega_i)\frac{|\cdot|^2}{2}$ with $\Bbb{P}(\omega_i=\pm 1)=1/2$ or $\exp(\beta\omega_i)\frac{|\cdot|^2}{2}$ with $\omega_i$ obeys a normal distribution. The interaction with the origin in the random field space is considered. Each time the field touches the origin, a reward $\epsilon\geq 0$ is given. Although these models are quite different from the pinning models studied in Giacomin (2007), the result about the gap between the annealed critical point and the quenched critical point stays the same.