John Noble, Matematisk Statistik
Linköpings Universitet


The Directed Polymer in a Random Environment

Abstract

Consider a random walk in Z, starting at the origin, and at each unit
time interval taking a step to the left, or to the right, each with
probability 1/2. After t steps, the mean squared displacement will be
t.

Now suppose that a random energies V(t,x), independent and identically
distributed with normal distribution (mean zero) are associated with
each point in Z_+ \times Z. Consider a change of measure whereby the
paths of the random walk are weighted according to the exponential of
their associated energy.

Under this change of measure, the mean squared displacement changes.
It has long been known numerically that for d=1, the mean squared
displacement is t^{4/3}. The process is superdiffusive.

In this seminar I outline a proof of the 4/3 superdiffusive exponent
for a related model, continuous in space and time, where the random
walk has been replaced by a Brownian motion and the energy is
described by a random field.

Taking the continuous limit simplifies the problem, but unfortunately
eliminates the low temperature phase that is present in the original
problem.