The project is devoted to the study of the time that simple random walk on 𝑍^𝑑 visits each
vertex. This is called local time profile, and it is pivotal in the study of processes obtained
by a change of measure of simple random walks, which we call “polymer measure”. Our
goal is to analyse localization of polymer measures. In particular we, review some old and
new results about polymers obtained by penalizing the paths of simple random walk by
their range. These “new” paths obey a shape theorem, i.e. the range will resemble balls in
𝑍^𝑑 of radius 𝑁^1/(𝑑+2) where 𝑁 is the time horizon.