## The Annals of Probability

- Ann. Probab.
- Volume 22, Number 1 (1994), 180-194.

### Domination Between Trees and Application to an Explosion Problem

Robin Pemantle and Yuval Peres

#### 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 $\sum f(n)^{-1} < \infty$.

#### Article information

**Source**

Ann. Probab., Volume 22, Number 1 (1994), 180-194.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176988855

**Digital Object Identifier**

doi:10.1214/aop/1176988855

**Mathematical Reviews number (MathSciNet)**

MR1258873

**Zentralblatt MATH identifier**

0806.60098

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Secondary: 60F15: Strong theorems 60E15: Inequalities; stochastic orderings

**Keywords**

First-passage percolation explosion tree domination

#### Citation

Pemantle, Robin; Peres, Yuval. Domination Between Trees and Application to an Explosion Problem. Ann. Probab. 22 (1994), no. 1, 180--194. doi:10.1214/aop/1176988855. https://projecteuclid.org/euclid.aop/1176988855