Exponential decay of connectivity and uniqueness in percolation on finite and infinite graphs

TL;DR

This paper establishes an upper bound for the uniqueness transition in percolation on graphs using spectral radii of non-backtracking matrices, showing exponential decay of connectivity under certain conditions.

Contribution

It introduces a spectral radius-based upper bound for the percolation transition and demonstrates exponential decay of connectivity with strong local connectivity assumptions.

Findings

Upper bound for the uniqueness transition in terms of spectral radii.

Exponential decay of connectivity for subgraphs below a critical threshold.

Connection between spectral properties and percolation behavior.

Abstract

We give an upper bound for the uniqueness transition on an arbitrary locally finite graph ${\cal G}$ in terms of the limit of the spectral radii $\rho\left[ H({\cal G}_t)\right]$ of the non-backtracking (Hashimoto) matrices for an increasing sequence of subgraphs ${\cal G}_t\subset{\cal G}_{t+1}$ which converge to ${\cal G}$. With the added assumption of strong local connectivity for the oriented line graph (OLG) of ${\cal G}$, connectivity on any finite subgraph ${\cal G}'\subset{\cal G}$ decays exponentially for $p<(\rho\left[ H({\cal G}^{\prime})\right])^{-1}$.