Gaps between roots in ballistic deposition on a strip: linear variance, Gaussian limits and a Poisson regime
Abstract
Mansour, Rastegar and Roitershtein studied the \emph{roots} of ballistic deposition on a cyclic strip of K sites, namely the sites that receive a particle at height one, and the number Di,K of pairs of consecutive roots at distance i+1. They computed the first two moments of Di,K for i≤7. They conjectured that for every i the mean and the variance are exactly linear in K, that Di,K satisfies a central limit theorem, and that the standardized vector (Di,K)i≥1 converges to a Gaussian process. We prove these conjectures. The key observation is that the roots are exactly the valleys of a uniformly random circular arrangement, so that Di,K counts occurrences of a family of consecutive patterns. We determine sizes from which the moments are exactly linear; the size for the mean is sharp for every i, and those for the second moments cannot be lowered in any case we checked. We give closed forms for the variance constants and for the limiting covariance; the mean constants agree with a known distribution of distances between surface peaks. The reduction and the exact moment formulas are formalized in Lean~4 with Mathlib. We also prove a Wasserstein bound of order (𝔼 Di,K)−1/2 that is uniform in i, and a local limit theorem. Finally we describe the limit completely when i grows with K: it is degenerate, Poisson or Gaussian according as 𝔼 Di,K tends to zero, to a positive constant, or to infinity.
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BibTeX
@misc{matijasevick2026gaps,
title = {{Gaps between roots in ballistic deposition on a strip: linear variance, Gaussian limits and a Poisson regime}},
author = {Mateo Matijasevick},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21094},
eprint = {2610.21094},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21094},
}
RIS
TY - UNPB
TI - Gaps between roots in ballistic deposition on a strip: linear variance, Gaussian limits and a Poisson regime
AU - Mateo Matijasevick
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21094
UR - https://arxiv.org/abs/2610.21094
AB - Mansour, Rastegar and Roitershtein studied the \emph{roots} of ballistic deposition on a cyclic strip of K sites, namely the sites that receive a particle at height one, and the number D_i,K of pairs of consecutive roots at distance i+1. They computed the first two moments of D_i,K for i≤7. They conjectured that for every i the mean and the variance are exactly linear in K, that D_i,K satisfies a central limit theorem, and that the standardized vector (D_i,K)_i≥1 converges to a Gaussian process. We prove these conjectures. The key observation is that the roots are exactly the valleys of a uniformly random circular arrangement, so that D_i,K counts occurrences of a family of consecutive patterns. We determine sizes from which the moments are exactly linear; the size for the mean is sharp for every i, and those for the second moments cannot be lowered in any case we checked. We give closed forms for the variance constants and for the limiting covariance; the mean constants agree with a known distribution of distances between surface peaks. The reduction and the exact moment formulas are formalized in Lean~4 with Mathlib. We also prove a Wasserstein bound of order (𝔼 D_i,K)^−1/2 that is uniform in i, and a local limit theorem. Finally we describe the limit completely when i grows with K: it is degenerate, Poisson or Gaussian according as 𝔼 D_i,K tends to zero, to a positive constant, or to infinity.
ER -