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arXiv · 1610.04897

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

Kathleen E. Hamilton·Leonid P. Pryadko
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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 $p<(\rho\left[ H({\cal G}^{\prime})\right])^{-1}$.

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Kathleen E. Hamilton, Leonid P. Pryadko. 2016-10-16. Exponential decay of connectivity and uniqueness in percolation on finite and infinite graphs. https://arxiv.org/abs/1610.04897

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