# Probability and Mathematical Physics Seminar

#### Restricted critical exponents in high-dimensional percolation

**Speaker:**
Jack Hanson, CUNY - City College

**Location:**
Warren Weaver Hall 512

**Date:**
Friday, February 1, 2019, 11 a.m.

**Synopsis:**

Critical percolation is fairly well-understood on \(Z^d\) for \(d > 11\). Exact values of many critical exponents are rigorously known: for instance, the “one-arm” probability that the origin is connected by an open path to distance \(r\) scales as \(r^{-2}\). However, most existing methods rely heavily on the symmetries of the lattice, so they do not extend to fractional spaces. We will discuss progress on these questions in the high-dimensional upper half-space (and within cubes), including the result that the half-space one-arm probability scales as \(r^{-3}\).