Loop vs. Bernoulli percolation on trees: strict inequality of critical values

Abstract

We study loop ensembles on locally finite rooted trees. The loops are induced by a Poisson process of links on the edges; the transposition-only case is the random interchange process, and the same framework covers loop representations of quantum spin systems. The set of edges carrying at least one untyped link forms i.i.d. Bernoulli bond percolation with retention probability $1 - \exp(-β)$, so an infinite loop can occur only if the corresponding link cluster is infinite. Our main results for Galton-Watson trees show both phenomena: if the offspring distribution has finite mean m in $(1,\infty)$, then, conditioned on survival, the loop threshold is strictly larger than the link threshold; in contrast, in the random interchange case a heavy-tail condition on the offspring distribution forces the loop and link thresholds, averaged over both the tree and the link configuration, to coincide at zero. The separation theorem follows from a stronger deterministic criterion based on local loop configurations that cut descendant subtrees out of link clusters.