Topics in Probability

October 24, 2017
3:00pm - 4:00pm
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 prove that a percolation-decorated RPM converges in law to SLE-decorated LQG in a certain topology. This is joint work with Bernardi and Sun. We then discuss works in progress with the goal of strengthening the topology of...

Fine Hall 110
Speaker: Nina Holden,
October 26, 2017
4:30pm - 5:30pm
Comparing exponential and Erdős–Rényi random graphs, and a general bound on the distance between Bernoulli random vectors

We present a general bound on the Wasserstein distance between two arbitrary distributions of sequences of Bernoulli random variables. The bound is in terms of 1) a mixing quantity for the Glauber dynamics of one of the sequences, and 2) a simple expectation of the other. The result is applied to estimate, with explicit error, expectations of functions of random vectors for some Ising models and exponential random graphs in "high temperature" regimes. Joint work with Gesine Reinert.

Fine Hall 601
Speaker: Nathan Ross,
University of Melbourne