Positive association of the oriented percolation cluster in randomly oriented graphs
| Autor: | Bienvenu, François |
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| Rok vydání: | 2017 |
| Předmět: | |
| Zdroj: | Combinatorics, Probability and Computing, 28(6):811-815 (2019) |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1017/S0963548319000191 |
| Popis: | Consider any fixed graph whose edges have been randomly and independently oriented, and write $\{S \leadsto i\}$ to indicate that there is an oriented path going from a vertex $s \in S$ to vertex $i$. Narayanan (2016) proved that for any set $S$ and any two vertices $i$ and $j$, $\{S \leadsto i\}$ and $\{S \leadsto j\}$ are positively correlated. His proof relies on the Ahlswede-Daykin inequality, a rather advanced tool of probabilistic combinatorics. In this short note, I give an elementary proof of the following, stronger result: writing $V$ for the vertex set of the graph, for any source set $S$, the events $\{S \leadsto i\}$, $i \in V$, are positively associated -- meaning that the expectation of the product of increasing functionals of the family $\{S \leadsto i\}$ for $i \in V$ is greater than the product of their expectations. Comment: This is the accepted (post-print) version. The example of application was removed |
| Databáze: | arXiv |
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