Multi-excited random walks on general trees
Abstract
We study multi-excited random walks on infinite locally finite trees. The model is described by placing a stack of ``cookies'' at each vertex, with one cookie consumed on each successive visit. As long as cookies remain, the walk is in an excited regime and makes nearest-neighbour steps according to a prescribed sequence of biases. Once the stack is exhausted, the walk is in a non-excited regime and moves uniformly among the neighbouring vertices. The model is closely connected to foraging ecology, where repeated visits to a location can deplete local resources and thereby influence the attractiveness of that location for future visits. We prove that the process exhibits a sharp phase transition between recurrence and transience. The critical threshold is determined by the branching-ruin number of the tree, which is equal to the Hausdorff dimension of the tree's boundary under a suitable metric. Our result substantially extends Zerner's recurrence-transience criterion for multi-excited random walks on ℤ [Probab. Theory Related Fields 133 (2005)] to general trees.
Related
- Branching Random Walks on Multi-Type Galton–Watson Trees: A Multifractal Analysis
- Shape theorem for excited random walks on the integer lattice
- Absorption probabilities of random walks without symmetry or general position
- Spectral Duality and Reset-Neutral Distributions in Random Walks with Multi-Site Geometric Resetting
- Limit shapes for centrally excited walks
BibTeX
@misc{mills2026multiexcited,
title = {{Multi-excited random walks on general trees}},
author = {Jack Mills and Tuan-Minh Nguyen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40084},
eprint = {2610.40084},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40084},
}
RIS
TY - UNPB
TI - Multi-excited random walks on general trees
AU - Jack Mills
AU - Tuan-Minh Nguyen
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40084
UR - https://arxiv.org/abs/2610.40084
AB - We study multi-excited random walks on infinite locally finite trees. The model is described by placing a stack of ``cookies'' at each vertex, with one cookie consumed on each successive visit. As long as cookies remain, the walk is in an excited regime and makes nearest-neighbour steps according to a prescribed sequence of biases. Once the stack is exhausted, the walk is in a non-excited regime and moves uniformly among the neighbouring vertices. The model is closely connected to foraging ecology, where repeated visits to a location can deplete local resources and thereby influence the attractiveness of that location for future visits. We prove that the process exhibits a sharp phase transition between recurrence and transience. The critical threshold is determined by the branching-ruin number of the tree, which is equal to the Hausdorff dimension of the tree's boundary under a suitable metric. Our result substantially extends Zerner's recurrence-transience criterion for multi-excited random walks on ℤ [Probab. Theory Related Fields 133 (2005)] to general trees.
ER -