Abstract

Percolation and Bunkbed Graphs

Nick Weininger
Mon Sep. 13 at 1:10pm in the Graduate Student Lounge

A bunkbed graph is the product of an arbitrary graph $G$ with the single edge $K_2$. A natural, and surprisingly difficult, conjecture states that for certain kinds of percolation on bunkbed graphs, the existence of a path from $(a,0)$ to $(b,0)$ is more likely than from $(a,0)$ to $(b,1)$, where $a,b$ are vertices of $G$. Recently, H\"aggstr\"om gave a beautiful short proof of a slight variation on this conjecture, using the random-cluster coupling technique from statistical physics. I will give H\"aggstr\"om's proof and discuss why its approach seems to be inapplicable to the original conjecture, even though the original intuitively ought to be much easier. I will also discuss other analogous results involving random processes on bunkbed graphs.