Probability Seminar

Topic #1 (Kavvadias): We consider the Schramm-Loewner evolution (SLE_{kappa}) for kappa in (4,8), which is the regime that the curve is self-intersecting but not space-filling. We let K be the set of kappa in (4,8) for which the adjacency graph of connected components of the complement of an SLE_{kappa} is almost surely connected, in the sense that for every pair of complementary components U,V there exist complementary components U_1,...,U_n with U_1 = U, U_n = V, and the boundaries of U_i and U_{i+1} intersect for each i between 1 and n-1. We show that the range of an SLE_{kappa} for kappa in K is almost surely conformally removable, which answers a question of Sheffield. As a step in the proof, we construct the canonical conformally covariant volume measure on the cut points of an SLE_{kappa} for kappa in (4,8) and establish a precise upper bound on the measure that it assigns to any Borel set in terms of its diameter. This is joint work with Jason Miller and Lukas Schoug.

Topic #2 (Narayanan): 

Hives, as defined by Knutson and Tao, are discrete concave functions on a triangular grid on an equilateral triangle of side n. It is known through the work of Knutson and Tao that the probability distribution of the spectrum of the sum of two independent random matrices with unitarily invariant distributions and given spectra can be expressed in terms of certain marginals of random hives.

We prove the existence of a surface tension function sigma depending on the Hessian, for continuum limits of random hives, and prove a large deviation principle for the large n limit of random hives in terms of sigma. Through the aforementioned connection, we also obtain a large deviation principle for the spectrum of the sum of two random matrices with given spectra.

This is joint work with Scott Sheffield.