Abstract

A polymer is a long chain of small molecules that is traditionally modelized using a Gibbs tranformation of the simple random walk law on $ZZ^d$, which we call the polymer measure. The Gibbs weight favors configurations that minimize a Hamiltonian, which reflects how the polymer interacts with its environement. We are interested in the typical configurations of polymers when its length goes to infinity. I will be focusing on a polymer placed in an averaged inhomogeneous Bernoulli-type percolation in $\ZZ$, which can be seen as a random walk ”randomly penalized” by its range, with a random cost for each site in the range. When the energy cost is i.i.d. at each site, the location of the edges of the polymer at di↵erent scales are determined through random variational problems. Then, I will explain how we hope to study a polymer in the random interlacement using recent works by Ding & Xu on the walk killed by Bernoulli percolation and works by van den Berg, Bolthausen & den Hollander on the Wiener sausage.