Abstract: We consider two models for bio-polymers, the gradient interaction and the Lacplacian one,
both with the Gaussian potential in the random environment. A random field φ : { 0, 1, ..., N } → R^d represents the position of the polymer path. The law of the field is given by exp( − ∑ i |∇ φi |^2 /2) where ∇ is the discrete gradient, and by exp( − ∑ i | ∆φi |^2 /2) where ∆ is the discrete Laplacian. For every Gaussian potential |·|^2 /2, a random charge is added as a factor: (1+βωi) |·|^2 /2 with P (ωi = ± 1) = 1/2 or exp(βωi) |·|^2 /2 with ωi 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 ϵ ≥ 0 is given. Although these models are quite different from the pinning models studied in G. Giacomin (2011), the result about the gap between the annealed critical point and the quenched critical point stays the same.