Gaps between roots in ballistic deposition on a strip: linear variance, Gaussian limits and a Poisson regime
Year: 2026
Type: preprint
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 $D_{i,K}$ of pairs of consecutive roots at distance $i+1$. They computed the first two moments of $D_{i,K}$ for $i\le7$. 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 $(D_{i,K})_{i\ge1}$ 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 $(\mathbb{E} D_{i,K})^{-1/2}$ 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 $\mathbb{E} D_{i,K}$ tends to zero, to a positive constant, or to infinity.
Source: arXiv (Cornell University)
Author: Matijasevick, Mateo
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222618352