This monograph contains the notes of lectures I gave in Saint Flour Probability Summer
School in July 2016. The two other courses were given by Paul Bourgade and by Scott …

Probabilistic models of directed polymers in random environment have received
considerable attention in recent years. Much of this attention has focused on integrable …

The first goal of this paper is to prove multiple asymptotic results for a time‐discrete and
space‐continuous polymer model of a random walk in a random potential. These results …

We study the directed last-passage percolation model on the planar square lattice with
nearest-neighbor steps and general iid weights on the vertices, outside of the class of …

These lecture notes discuss several related features of the exactly solvable two-dimensional
corner growth model with exponentially distributed weights. A key property of this model is …

We discuss variational formulas for the law of large numbers limits of certain models of
motion in a random medium: namely, the limiting time constant for last-passage percolation …

We develop ergodic theory of the inviscid Burgers equation with random kick forcing in
noncompact setting. The results are parallel to those in our recent work on the Burgers …

We introduce new probabilistic arguments to derive optimal-order central moment bounds in
planar directed last-passage percolation. Our technique is based on couplings with the …

Directed polymers in random environment have usually been constructed with a simple
random walk on the integer lattice. It has been observed before that several standard results …

We consider a space-continuous and time-discrete polymer model for positive temperature
and the associated zero temperature model of last passage percolation type. In our previous …