Absorption probabilities of random walks without symmetry or general position
Abstract
Let S1,S2,… be a random walk in ℝd with independent identically distributed increments. We show that, for every distribution of the increments and for every n≥1, ℙ[0∉conv(S1,…,Sn)] = ∑m=1n1/m!∑substackk1,…,km≥1 k1+…+km=n1/(k1⋯ km) ℙ[0∉conv(S(1)k1,…,S(m)km)], where S(1),S(2),… are independent copies of the walk. No symmetry, general position or moment assumptions are needed. The result rests on a deterministic identity: for arbitrary vectors x1,…,xn, the number of orderings for which the convex hull of the partial sums avoids the origin equals the number of permutations for which the convex hull of the cycle sums avoids it. We derive it from a permutation-averaging identity for valuations obtained in our earlier work. Consequences include the identities of Sparre Andersen and Spitzer, the distribution-free absorption probabilities of Kabluchko, Vysotsky and Zaporozhets, their generating function found by Petrov, Randon-Furling and Zaporozhets, and a formula for the probability that the origin is never absorbed.
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BibTeX
@misc{bolotin2026absorption,
title = {{Absorption probabilities of random walks without symmetry or general position}},
author = {Artyom Bolotin and Dmitry Zaporozhets},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33216},
eprint = {2610.33216},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33216},
}
RIS
TY - UNPB
TI - Absorption probabilities of random walks without symmetry or general position
AU - Artyom Bolotin
AU - Dmitry Zaporozhets
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.33216
UR - https://arxiv.org/abs/2610.33216
AB - Let S_1,S_2,… be a random walk in ℝ^d with independent identically distributed increments. We show that, for every distribution of the increments and for every n≥1, ℙ[0∉conv(S_1,…,S_n)] = ∑_m=1^n1/m!∑_substackk_1,…,k_m≥1 k_1+…+k_m=n1/(k_1⋯ k_m) ℙ[0∉conv(S^(1)_k_1,…,S^(m)_k_m)], where S^(1),S^(2),… are independent copies of the walk. No symmetry, general position or moment assumptions are needed. The result rests on a deterministic identity: for arbitrary vectors x_1,…,x_n, the number of orderings for which the convex hull of the partial sums avoids the origin equals the number of permutations for which the convex hull of the cycle sums avoids it. We derive it from a permutation-averaging identity for valuations obtained in our earlier work. Consequences include the identities of Sparre Andersen and Spitzer, the distribution-free absorption probabilities of Kabluchko, Vysotsky and Zaporozhets, their generating function found by Petrov, Randon-Furling and Zaporozhets, and a formula for the probability that the origin is never absorbed.
ER -