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

Abstract

We give an upper bound for the uniqueness transition on an arbitrary locally finite graph $\mathcal{G}$ in terms of the limit of the spectral radii $\rho \left[H({\mathcal{G}}_t)\right]$ of the non-backtracking (Hashimoto) matrices for an increasing sequence of subgraphs ${\mathcal{G}}_t\subset {\mathcal{G}}_{t+1}$ which converge to $\mathcal{G}$. With the added assumption of strong local connectivity for the oriented line graph (OLG) of $\mathcal{G}$, connectivity on any finite subgraph ${\mathcal{G}}^{\prime }\subset \mathcal{G}$ decays exponentially for $p<(\rho \left[H({\mathcal{G}}^{\prime })\right]{)}^{-1}$.

2 pages. Abstract for the SIAM Workshop on Network Science (NS16), July 15-16, 2016, Boston, Massachusetts

References in corpus (1)

• Spectral bounds for percolation on directed and undirected graphs