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Accessible Percolation with Crossing Valleys on $n$-ary Trees. (arXiv:1712.04548v1 [math.PR])
来源于:arXiv
In this paper, also motivated by evolutionary biology and evolutionary
computation, we study a variation of the accessibility percolation model.
Consider a tree whose vertices are labeled with random numbers. We study the
probability of having a monotone subsequence of a path from the root to a
leave, where any $k$ consecutive vertices in the path contain at least one
vertex of the subsequence. An $n$-ary tree, with height $h$, is a tree whose
vertices at distance at most $h-1$ to the root have $n$ children. For the case
of $n$-ary trees, we proof that, as $h$ tends to infinity the probability of
having such subsequence: tends to 1, if $n(h)\geq c\sqrt[k]{h/(ek)} $ and
$c>1$; and tends to 0, if $n(h)\leq c\sqrt[k]{h/(ek)} $ and $c<1$. 查看全文>>