Random conformally invariant curves and quantum group techniques

Stochastic Analysis Seminar Series

In this talk we consider two questions about conformally invariant random curves known as Schramm-Loewner evolutions (SLE). The first question is about the "boundary zig-zags", i.e. the probabilities for a chordal SLE to pass through small neighbourhoods of given boundary points in a given order. The second question is that of obtaining explicit descriptions of "multiple SLE pure geometries", i.e. those extremal multiple SLE probability measures which cannot be expressed as non-trivial convex combinations of other multiple SLEs. For both problems one needs to find solutions of a system of partial differential equations with asymptotic conditions written recursively in terms of solution of the same problem with a smaller number of variables. We present a general correspondence, which translates these problems to linear systems of equations in finite dimensional representations of the quantum group U_q(sl_2), and we then explicitly solve these systems. The talk is based on joint works with Eveliina Peltola (Helsinki), and with Niko Jokela (Santiago de Compostela) and Matti Järvinen (Crete).


Kalle Kytola (Helsinki University of Technology)

Monday, May 13, 2013 - 15:45
to 16:45