Critical Parameters for Loop and Bernoulli Percolation

Abstract

We consider a class of random loop models (including the random interchange process) that are parametrised by a time parameter $\beta\geq 0$. Intuitively, larger $\beta$ means more randomness. In particular, at $\beta=0$ we start with loops of length 1 and as $\beta$ crosses a critical value $\beta_c$, infinite loops start to occur almost surely. Our random loop models admit a natural comparison to bond percolation with $p=1-e^{-\beta}$ on the same graph to obtain a lower bound on $\beta_c$. For those graphs of diverging vertex degree where $\beta_c$ and the critical parameter for percolation have been calculated explicitly, that inequality has been found to be an equality. In contrast, we show in this paper that for graphs of bounded degree the inequality is strict, i.e. we show existence of an interval of values of $\beta$ where there are no infinite loops, but infinite percolation clusters almost surely.