Brownian motion is a continuum scaling limit for a wide class of random processes, and
there has been great success in developing a theory for its properties (such as distribution …

Directed polymers in random environment can be thought of as a model of statistical
mechanics in which paths of stochastic processes interact with a quenched disorder …

Suppose we immerse a large porous stone in a bucket of water. What is the probability that
the centre of the stone is wetted? In formulating a simple stochastic model for such a …

We study a certain random growth model in two dimensions closely related to the one-
dimensional totally asymmetric exclusion process. The results show that the shape …

First-passage percolation (FPP) is a fundamental model in probability theory that has a wide
range of applications to other scientific areas (growth and infection in biology, optimization in …

Abstract We study a (1+ 1)-dimensional directed polymer in a random environment on the
integer lattice with log-gamma distributed weights. Among directed polymers, this model is …

Understanding the fluctuations of random objects is one of the major goals of probability
theory. There is a whole subfield of probability and analysis, called concentration of …

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 …

Let ̃ B (t) be the set of sites reached from the origin by time t in standard first-passage
percolation on Z d, and let B 0 (roughly lim ̃ B (t)/t) be its deterministic asymptotic shape. We …

Consider a realization of a Poisson process in ℝ 2 with intensity 1 and take a maximal
up/right path from the origin to (N, N) consisting of line segments between the points, where …