Critical curve of loop percolation on the $d$-regular tree

Abstract

We consider clusters formed by a Poisson ensemble of random walk loops on the $d$-regular tree with an intensity parameter $\hat{I}\pm >0$ and a killing parameter $\hat{I}º>-1$; the latter penalizes ($\hat{I}º>0$) or favors ($\hat{I}º<0$) the appearance of large loops. We obtain an implicit formula for the critical curve $\hat{I}º\mapsto \hat{I}{\pm }_c(\hat{I}º)$ for the percolation phase transition; the curve is positive if and only if $\hat{I}º>\hat{I}{º}_c=\frac{2\sqrt{d-1}}{d}-1$, differentiable away from $\hat{I}{º}_c$, and has order $\sqrt{\hat{I}º-\hat{I}{º}_c}$ as $\hat{I}º\downarrow \hat{I}{º}_c$ and order $(1+\hat{I}º{)}^2$ as $\hat{I}º\to \infty$. We show that for each $\hat{I}º>-1$, an infinite cluster exists exactly when $\hat{I}\pm >\hat{I}{\pm }_c(\hat{I}º)$. Finally, we identify the near-critical behavior of the susceptibility and the percolation probability: for $\hat{I}º>\hat{I}{º}_c$, the critical exponents take the mean-field values, while for $\hat{I}º=\hat{I}{º}_c$, the phase transition is of a higher order with the percolation probability decaying quadratically in $\hat{I}\pm -\hat{I}{\pm }_c$.

29 pages, 2 figures. Comments are welcome!