Quenched Survival of Bernoulli Percolation on Galton-Watson Trees
| Autor: | Michelen, Marcus, Pemantle, Robin, Rosenberg, Josh |
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| Rok vydání: | 2018 |
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| Druh dokumentu: | Working Paper |
| Popis: | We explore the survival function for percolation on Galton-Watson trees. Letting $g(T,p)$ represent the probability a tree $T$ survives Bernoulli percolation with parameter $p$, we establish several results about the behavior of the random function $g(\mathbf{T} , \cdot)$, where $\mathbf{T}$ is drawn from the Galton-Watson distribution. These include almost sure smoothness in the supercritical region; an expression for the $k\text{th}$-order Taylor expansion of $g(\mathbf{T} , \cdot)$ at criticality in terms of limits of martingales defined from $\mathbf{T}$ (this requires a moment condition depending on $k$); and a proof that the $k\text{th}$ order derivative extends continuously to the critical value. Each of these results is shown to hold for almost every Galton-Watson tree. Comment: 38 pages |
| Databáze: | arXiv |
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