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arXiv · 2107.14173

An upper bound for $p_c$ in range-$R$ bond percolation in two and three dimensions

Jieliang Hong
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Abstract

An upper bound for the critical probability of long range bond percolation in $d=2$ and $d=3$ is obtained by connecting the bond percolation with the SIR epidemic model, thus complementing the lower bound result in Frei and Perkins arXiv:arch-ive/1603.04130. A key ingredient is that we establish a uniform bound for the local times of branching random walk by calculating their exponential moments and by using the discrete versions of Tanaka's formula and Garsia's Lemma.

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Jieliang Hong. 2021-07-29. An upper bound for $p_c$ in range-$R$ bond percolation in two and three dimensions. https://arxiv.org/abs/2107.14173

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