SCIENTIFIC ACTIVITIES - Fields Institute - University of Toronto

Directed Polymers and Random Growth
JEREMY QUASTEL IS PROFESSOR AND ASSOCIATE CHAIR OF
mathematics at the University of Toronto. He was one of the delegates
representing Canada at the International Congress for Mathematicians in
Hyderabad this year (where he presented his work in Section 13: Probability
and Statistics) and is co-organizer of the Thematic Program on Dynamics and
Transport in Disordered Systems starting January 2011 at the Fields Institute.
His talk at the Fields Institute’s 2010 Annual General Meeting was
titled Directed polymers and random growth, a topic of probability theory and
stochastic differential equations. He spoke of the relationship between the
process of ballistic aggregation and discrete and continuum directed random
polymer models.
What does a random surface look like after a long period of time?
Ballistic aggregation is meant to answer this question, a discrete model in
one-dimension for blocks falling and building up on the 1D integer lattice.
The relevance of this theory to science is that understanding this process will
allow engineers to develop new tools to build materials by spraying atoms onto
a surface. Understanding of this and analogous models are one of the central
themes Quastel would like to see develop in his thematic program. He reports
that there have been a number of significant advances in the study of these
models and will certainly add excitement to the activity at Fields next year.
His lecture was split into two parts, background and recent results. The
directed random polymer models about which Quastel spoke were in the 2D
randomly weighted integer lattice. Each point in the lattice is represented by
W with an imposed random walk X that goes through the lattice collecting
i , j
i
the random values W and sums up the values. Taking the expectation
i , j
(summing all the possible paths and taking the logarithm of the sum) gives
us the free energy on the lattice. Physicists predict that the free energies of
the discrete random polymer model will give valuable information about the
object.
Quastel spoke about some reformulation of the GUE Tracy-Widom
models from random matrix theory in terms of the rescaled distribution of the
principle eigenvalue of a randomly chosen matrix from the GUE-TW. To his
surprise, these results are useful for the continuum directed polymer models.
In the context of the discrete model, Quastel mentioned his interest in
the behaviour emering from the strong coupling between the random walks
and the random lattice. The KPZ (Kardar-Parisi-Zhang) model governs
anything that experiences growth governed by randomness at different
sites, with added non-linearity. These models are extremely general. Quastel
mentioned his interest in having the theoretical tools to make robust
predictions over the KPZ universality class. In models studied in the past, as
the dimension increases, randomness behaviour undergoes a phase transition
1
3
where the n and Tracy-Widom distributions are replaced with normal
Gaussian behaviour and random walks.
The results obtained in work done with Gideon Amir and Ivan Corwin
are contained in a recent paper concerning the continuum directed random
polymer. The paper contains all the necessary formulas to construct a
distribution for the model. Quastel stated some results about solutions that
locally look like Brownian Motion, which start with smooth initial conditions.
Motivation for this problem comes from results in the field of liquid crystal
turbulence in which experiments were carried out in December 2009.
‘Jeremy Quastel’ continued on page 20
LECTURES
FIELDS INSTITUTE Research in Mathematical Sciences | FIELDSNOTES 13