Sharp transition in self-avoiding walk on random conductors on a tree

TL;DR

This paper investigates the phase transition behavior of self-avoiding walks on trees with random conductances, revealing a sharp transition between regimes where quenched and annealed critical points coincide or differ.

Contribution

It provides a new heuristic approach using fractional moment estimates to analyze the critical points in different disorder regimes.

Findings

Quenched and annealed critical points coincide in the weak disorder regime.

Critical points differ in the strong disorder regime.

The paper offers an alternative heuristic method for critical point estimation.

Abstract

We consider self-avoiding walk on a tree with random conductances. It is proven that in the weak disorder regime, the quenched critical point is equal to the annealed one, and that in the strong disorder regime, these critical points are strictly different. Derrida and Spohn, and Baffet, Patrick and Pul$\acute{\rm e}$ give the exact value of the quenched critical point. We give another heuristic approach by the fractional moment estimate.