Infinite-mean Galton-Watson trees have zero random-interchange threshold

Source abstract

We prove that, on a Galton-Watson tree with almost surely finite offspring and infinite offspring mean, the random interchange process has critical parameter zero for infinite cycles, conditionally on survival. Combined with the finite-mean strict inequality established in \emph{arXiv:2503.03319}, this gives a dichotomy for locally finite Galton-Watson trees with offspring mean in $(1,\infty]$: the loop and link critical parameters coincide exactly in the infinite-mean case. The proof reduces the tree problem to a finite permutation estimate derived from peeling estimates for random surfaces.