Abstract

A polymer in a random environment is a Gibbs transformation of the simple random walk (SRW) on $ZZ^d$, whose Hamiltonian reflects how it interacts with its environement. The environment can be a fixed realization (quenched) or averaged (annealed). I will be focusing on an annealed one-dimensional polymer placed in an (in)homogeneous Bernoulli percolation, which can be seen as a SRW randomly penalized by its range. When the Bernoulli parameter are i.i.d. at each site, the location of the edges of the (now quenched) polymer at different scales are determinde through a random variational problem. Then, I will explain some work in progress for a polymer in the vacant set of the random interlacements, notably for the annealed model which is related to the SRW penalized by its capacity. This leads to investigate the capacity of the SRW constrained to stay in a small ball, as well as the capacity of a vanishing ranom interlacement intersected with a large ball.