arXiv 0909.40372009-09math.PRmath.CO

Random induced subgraphs of Cayley graphs induced by transpositions

Emma Y. Jin, Christian M. Reidys

备注：: 18 pages, 1 figure

英文摘要：

In this paper we study random induced subgraphs of Cayley graphs of the symmetric group induced by an arbitrary minimal generating set of transpositions. A random induced subgraph of this Cayley graph is obtained by selecting permutations with independent probability, $λ_n$. Our main result is that for any minimal generating set of transpositions, for probabilities $λ_n=\frac{1+ε_n}{n-1}$ where $n^{-{1/3}+δ}\le ε_n<1$ and $δ>0$, a random induced subgraph has a.s. a unique largest component of size $\wp(ε_n)\frac{1+ε_n}{n-1}n!$, where $\wp(ε_n)$ is the survival probability of a specific branching process.

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