Kytölä abstract

Kalle Kytölä: Boundary zig-zags of random conformally invariant curves

ABSTRACT:
Schramm-Loewner evolutions (SLE) are conformally invariant random curves that describe scaling limits of interfaces in various models of critical statistical physics in two dimensions. In this talk we consider "boundary zig-zags", i.e. the probabilities for such curves to pass through small neighborhoods of given boundary points in a given order. We find formulas for these probability amplitudes by solving a system of partial differential equations with asymptotics requirements written recursively in terms of solution of the same problem with a smaller number of variables. The solution is then based on a general correspondence, which translates such problems to linear systems of equations in finite dimensional representations of the quantum group U_q(sl_2). The talk is based on joint works with Niko Jokela (Santiago de Compostela) and Matti Järvinen (Crete), with Eveliina Peltola (Helsinki), and on some ongoing work with Konstantin Izyurov (Helsinki).