A Heteropolymer in a Medium with Random Droplets

1.1. Definition of the model. We start with the definition of the model. The dimension is denoted by d+1, d≥ 1. Then our model has three random ingredients: (1) a (d+ 1)-dimensional random medium, described by η; (2) a (d+1)-dimensional directed random walk, described by S; (3) a sequence of random variables ω, taking values ±1 describing the random monomers. So let us formally define all these objects. 1. Random medium. Choose η = (ηi)i∈Z: an i.i.d. sequence of random variables taking the value 1 with probability p ∈ [0,1] and taking value −1 with probability 1− p. Then the random medium is defined as follows: