Stony Brook Analysis Seminar, Fall 2017
Thursday 2.30 - 3.30 pm
Nina Holden (MIT)
Convergence of percolation-decorated triangulations to SLE and LQG
The Schramm-Loewner evolution (SLE) is a family of random fractal curves, which is the proven or conjectured scaling limit of a variety of two-dimensional lattice models in statistical mechanics. Liouville quantum gravity (LQG) is a model for a random surface which is the proven or conjectured scaling limit of discrete surfaces known as random planar maps (RPM). We consider scaling limits results for percolation-decorated RPM to SLE-decorated LQG. Based on joint works with Bernardi, Garban, Gwynne, Miller, Sepulveda, Sheffield, and Sun.
Slides from the talk are available here.
Megan Bernstein (Georgia Tech)
Progress in showing cutoff for random walks on the symmetric group
Abstract. Cutoff is a remarkable property of many Markov chains in which they rapidly transition from an unmixed to a mixed distribution. Most random walks on the symmetric group, also known as card shuffles, are believed to mix with cutoff, but we are far from being able to proof this. We will survey existing cutoff results and techniques for random walks on the symmetric group, and present three recent results: cutoff for a biased transposition walk, cutoff for the random-to-random card shuffle (answering a 2001 conjecture of Diaconis), and pre-cutoff for the involution walk, generated by permutations with a binomially distributed number of two-cycles. The results use either probabilistic techniques such as strong stationary times or diagonalization through algebraic combinatorics and representation theory of the symmetric group.
Includes joint work with Nayantara Bhatnagar, Evita Nestoridi, and Igor Pak.
Paul Bourgade (Courant Institute)
Random matrices, the Riemann zeta function and trees
Abstract. Fyodorov, Hiary & Keating have conjectured that the maximum of the characteristic polynomial of random unitary matrices behaves like extremes of log-correlated Gaussian fields. This allowed them to predict the typical size of local maxima of the Riemann zeta function along the critical axis. I will first explain the origins of this conjecture, and then outline the proof for the leading order of the maximum, for unitary matrices and the zeta function. This talk is based on joint works with Arguin, Belius, Radziwill and Soundararajan.
Frank Thorne (South Carolina)
Max Engelstein (MIT)