A shape theorem for a radially excited random walk in dimensions d >= 2
Abstract
NOTE (October 2026): This paper has been superseded by the joint paper "Limit shapes for centrally excited walks", by Ahmed Bou-Rabee, Tuan-Minh Nguyen, Yuval Peres and Arvind Singh (arXiv:2610.10162), which brings together the results and ideas of three independent drafts. We consider a once-excited random walk on ℤd, d ≥ 2, where the walk on its first visit has a bias of constant strength β>0 toward the origin and moves like a simple symmetric random walk on subsequent visits to that site. We show that the walk is recurrent and prove an almost sure spherical shape theorem: the trace after n steps is asymptotically a Euclidean ball centered at the origin, with radius proportional to n1/(d+1), and the local times have an asymptotically conical profile. This result confirms a conjecture of Kozma (2007) on the shape of the trace for this model.
Related
- Shape theorem for excited random walks on the integer lattice
- Optimal central limit theorem for bounded random variables in high dimensions
- Quantitative Estimates of the Field Excited by an Emitter in the Narrow Region Between Two Strictly Convex Inclusions in Two Dimensions
- Recurrence and range of the balanced excited random walk M(2,1,2)
- Multi-excited random walks on general trees
BibTeX
@misc{singh2026shape,
title = {{A shape theorem for a radially excited random walk in dimensions d >= 2}},
author = {Arvind Singh},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12138},
eprint = {2610.12138},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05788575v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A shape theorem for a radially excited random walk in dimensions d >= 2
AU - Arvind Singh
PY - 2026
DA - 2026/10/08
PB - HAL
DO - 10.48550/arxiv.2610.12138
UR - https://hal.science/hal-05788575v1
AB - NOTE (October 2026): This paper has been superseded by the joint paper "Limit shapes for centrally excited walks", by Ahmed Bou-Rabee, Tuan-Minh Nguyen, Yuval Peres and Arvind Singh (arXiv:2610.10162), which brings together the results and ideas of three independent drafts. We consider a once-excited random walk on ℤ^d, d ≥ 2, where the walk on its first visit has a bias of constant strength β>0 toward the origin and moves like a simple symmetric random walk on subsequent visits to that site. We show that the walk is recurrent and prove an almost sure spherical shape theorem: the trace after n steps is asymptotically a Euclidean ball centered at the origin, with radius proportional to n^1/(d+1), and the local times have an asymptotically conical profile. This result confirms a conjecture of Kozma (2007) on the shape of the trace for this model.
ER -