Bond Percolation in Frustrated Systems

Abstract

We study occurrence and properties of percolation of occupied bonds in systems withrandom interactions and, hence, frustration.We develop a general argument, somewhat like Peierls’ argument, by which we show that in $\mathbb{Z},d\geq2$, percolation occurs for all possible interactions (provided they are bounded away from zero) if the parameter $p \in 1$, regulating the density of occupied bonds, is high enough. If the interactions are i.i.d. random variables then we determine bounds on the values of $p$ for which percolation occurs for all, almost all but not all, almost none but some, or none of the interactions. Motivations of this work come from the rigorous analysis of phase transitions in frustrated statistical mechanics systems.