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 ∈ (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 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.
1 Bienaymé trees are also known as Galton-Watson trees. See [1] for a quick discussion on those terminologies. 1
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BibTeX
@misc{aurillard2026subordination,
title = {{Subordination of discrete snakes}},
author = {Antoine Aurillard and Mathieu Mourichoux},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02115},
eprint = {2610.02115},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05789911v1},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Subordination of discrete snakes
AU - Antoine Aurillard
AU - Mathieu Mourichoux
PY - 2026
DA - 2026/10/01
PB - HAL
DO - 10.48550/arxiv.2610.02115
UR - https://hal.science/hal-05789911v1
AB - 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 ∈ (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 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. 1 Bienaymé trees are also known as Galton-Watson trees. See [1] for a quick discussion on those terminologies. 1
ER -