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

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 $p<(\rho\left[ H({\cal G}^{\prime})\right])^{-1}$.