Subordination of discrete snakes

Abstract

Motivated by applications in random geometry, we investigate the notion of subordination of snakes in the discrete setup. More precisely, given a random walk $W$ indexed by a tree $T$ and with steps in $\{...,-1,0,1\}$, we consider its subordinate tree obtained by contracting every edge of $T$ that does not lead to a new record of the walk $W$. When the underlying tree $T$ is a Bienaymé-Galton-Watson tree, we characterize the distribution of this subordinate tree. In particular, when $T$ has a critical offspring distribution in an $α$-stable domain of attraction with $α\in(1,2]$, and under a light tails assumption on the steps, we prove that the associated subordinate tree is itself a Bienaymé-Galton-Watson tree with an offspring distribution in an $\frac{α+1}{2}$-stable domain of attraction. Along the way, we obtain the asymptotic tail of the maximal displacement of the critical branching random walk $W$ in this stable regime, under minimal assumptions. Finally, we use these results to prove scaling limit statements about the subordinate tree.