# 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 $$r^{-3}$$.