IMI Interdisciplinary Mathematics InstituteCollege of Arts and Sciences

## Taxi walks and the hard-core distribution on $\mathbb{Z}$2

• March 17, 2017
• 2:30 p.m.
• LeConte 312

## Abstract

The $\textit{hard-core}$ distribution on a graph $G$ is the probability distribution on the independent sets of $G$ (sets of mutually non-adjacent vertices) in which each such set $I$ has probability proportional to $\lambda^{|I|}$, for some $\lambda > 0$.

The hard-core distribution ar​i​se​s​ as a simple model of the occupation of space by a gas with massive particles, and is mainly of interest because it has the potential to exhibit a ​gas-solid phase transition: for small $\lambda$ a typical configuration should be a mostly uncorrelated sparse set of vertices, while for larger $\lambda$ it should be a highly correlated dense subset of a maximum independent set.

I'll focus on the integer lattice $\mathbb{Z}^2$, where we strongly expect a transition point to exist. I'll discuss recent work with Blanca, ​Chen, ​Randall and Tetali, where we show that the solid phase can be better understood by introducing a new class of self-avoiding walks on ${\mathbb Z}^2$ that mimics the movement of taxi cabs around Manhattan.

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