Recent work on Stochastic Loewner Evolution (SLE) by Lawler, Schramm and Werner has established a connection between SLE(6), critical percolation and a special case of reflected Brownian motion. This result motivated us to study numerically a model known as the self-avoiding trail. In a naive sense, this model interpolates between Brownian motion and the exploration process of critical percolation, and it is natural to believe that this model should also be closely related to reflected Brownian motion. In this lecture, I will provide background on reflected Brownian motions, and present our numerical results on the relation between reflected Brownian motion and the self-avoiding trail.
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 (impurities), depending on both time and space.
Two related quantities are used to describe how strongly the polymer feel the medium: the limit of some natural martingale, and a quenched Lyapunov exponent.
We will discuss the following:
(i) for large dimension and high temperature, the polymer is diffusive
(ii) for small dimension the above limit vanishes
(iii) strict positivity of the exponent implies localization of the paths (this case being typical of low temperature)