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

TL;DR

The paper addresses whether a phase transition in loop percolation on random trees implies a corresponding transition in the underlying bond percolation, and shows a strict separation of thresholds: $\beta_c^{\text{loop}} > \beta_c^{\text{link}}$ for a broad class of trees, including Galton–Watson trees with finite mean offspring. It develops a robust coupling framework to compare loop percolation with inhomogeneous Bernoulli percolation via pruning and delayed-pruning mechanisms, and uses potential theory on trees to connect thresholds to capacities and harmonic measures. For Galton–Watson trees, it provides a sharp criterion: the inequality holds exactly when $\mathbb{E}[Z]=\sum_k k\zeta_k \in (1,\infty)$, linking the phase transition to offspring distribution tails. These results illuminate phase transitions on sparse graphs and supply tools for analyzing loop representations of quantum spin systems on random trees.

Abstract

We consider loop ensembles on random trees. The loops are induced by a Poisson process of links sampled on the underlying tree interpreted as a metric graph. We allow two types of links, crosses and double bars. The crosses-only case corresponds to the random-interchange process, the inclusion of double bars is motivated by representations of models arising in mathematical physics. For a large class of random trees, including all Galton-Watson trees with mean offspring number in $(1,\infty)$, we show that the threshold Poisson intensity at which an infinite loop arises is strictly larger than the corresponding quantity for percolation of the untyped links. The latter model is equivalent to i.i.d. Bernoulli bond percolation. An important ingredient in our argument is a sensitivity result for the bond percolation threshold under downward perturbation of the underlying tree by a general finite range percolation. We also provide a partial converse to the strict-inequality result in the case of Galton-Watson trees and improve previously established criteria for the existence of an infinite loop in the case were the Galton-Watson tree has Poisson offspring.

Figures (2)

• Figure 1: Example loop configurations. On the left is and example of the case $u=1$ of only crosses. On the right the case $u\in(0,1)$ so that crosses and bars are both present.
• Figure 2: Top: examples of an edge ($\{2,3\}$) and three link configurations so that the edge is pruning, blocking but not pruning, and neither pruning nor blocking, respectively. Bottom left: the effect of a pruning edge $e$ on the path of $L_o$ in $T$. Bottom right: the effect of a blocking edge $e$ on the path of $L_o$ in $T$.

Theorems & Definitions (24)

• Remark 2.1
• Example 2.2
• Definition 3.1: Pruning percolation
• Definition 3.2: Pruning edges
• Remark 3.3
• Lemma 3.4
• proof
• Proposition 3.5
• proof
• Theorem 4.1