### Critical Parameters for Loop and Bernoulli Percolation

Probability Seminar

18th October 2019, 3:30 pm – 4:30 pm

Fry Building, LG.22

We consider a class of random loop models (including the random interchange process) that are parametrised by a time parameter $\beta\geq 0$. 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. These random loop models admit a natural comparison to bond percolation 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 here 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.

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