Domination Between Trees and Application to an Explosion Problem

Abstract

We define a notion of stochastic domination between trees, where one tree dominates another if, when the vertices of each are labeled with independent, identically distributed random variables, one tree is always more likely to contain a path with a specified property. Sufficient conditions for this kind of domination are (1) more symmetry and (2) earlier branching. We apply these conditions to the problem of determining how fast a tree must grow before first-passage percolation on the tree exhibits an explosion, that is to say, infinitely many vertices are reached in finite time. For a tree in which each vertex at distance $n - 1$ from the root has $f(n)$ offspring, $f$ nondecreasing, an explosion occurs with exponentially distributed passage times if and only if $\sum f(n)^{-1} < \infty$.