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

We consider clusters formed by a Poisson ensemble of random walk loops on the $d$-regular tree with an intensity parameter $\alpha>0$ and a killing parameter $\kappa>-1$; the latter penalizes ($\kappa > 0$) or favors ($\kappa <0$) the appearance of large loops.

We obtain an implicit formula for the critical curve $\kappa\mapsto \alpha_c(\kappa)$ for the percolation phase transition; the curve is positive if and only if $\kappa>\kappa_c = \frac{2\sqrt{d-1}}{d}-1$, differentiable away from $\kappa_c$, and has order $\sqrt{\kappa-\kappa_c}$ as $\kappa\downarrow\kappa_c$ and order $(1+\kappa)^2$ as $\kappa\to\infty$.

We show that for each $\kappa>-1$, an infinite cluster exists exactly when $\alpha>\alpha_c(\kappa)$.

Finally, we identify the near-critical behavior of the susceptibility and the percolation probability: for $\kappa>\kappa_c$, the critical exponents take the mean-field values, while for $\kappa=\kappa_c$, the phase transition is of a higher order with the percolation probability decaying quadratically in $\alpha-\alpha_c$.