Anthony Metcalfe: Random matrices, random tilings, random interlaced interlaced configurations, and universality

Title: Random matrices, random tilings, random interlaced interlaced configurations, and universality.

Abstract: In this talk I will give an overview of my research over the last few years concerning
universal asymptotic behaviours of Gelfand-Tsetlin patterns (GT-patterns). A GT-pattern of
depth n in a configuration of particles arranged in n rows/levels. There are n particles on
the row n (top level), n-1 particles on row n-1, n-2 particles on row n-2, etc. The particles
also interlace in the following sense: For any r>1, there is exactly 1 particle on row r-1
between any two consecutive particles on row r.

Consider the uniform probability measure on the set GT-patterns of depth n with the
particles on row n in deterministic positions. Call this the discrete process when the particle
positions on each row are restricted to integer values, and the continuous process when
the particles positions can take any real-value. The continuous process equivalently describes
the eigenvalue minor process of an ensemble of random Hermitian matrices, and the discrete
process equivalently describes random tilings of `half-hexagons' with lozenges.

Let n increase under the assumption that the empirical distribution of the particles on the
top row converges weakly, and these particles otherwise satisfy some mild regulatory restrictions.
We will explore the asymptotic shape of the resulting GT-patterns, and the universal local
asymptotic behaviours in neighbourhoods of the asymptotic shape.