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

Kathleen E. Hamilton, Leonid P. Pryadko
2016 arXiv   pre-print
We give an upper bound for the uniqueness transition on an arbitrary locally finite graph G in terms of the limit of the spectral radii ρ[ H( G_t)] of the non-backtracking (Hashimoto) matrices for an increasing sequence of subgraphs G_t⊂ G_t+1 which converge to G. With the added assumption of strong local connectivity for the oriented line graph (OLG) of G, connectivity on any finite subgraph G'⊂ G decays exponentially for p<(ρ[ H( G^')])^-1.
arXiv:1610.04897v1 fatcat:tnfdvilumbdvhdhazbbek3y5ca