Large Deviations in One-Dimensional Random Sequential Adsorption

TL;DR

This paper studies a class of one-dimensional lattice RSA models, deriving full counting statistics and variance of occupation numbers, with exact solutions for the minimal case and a perturbation method for general cases.

Contribution

It introduces a solvable Riccati equation for the minimal model and a perturbation approach to compute cumulants for all models.

Findings

Full counting statistics computed for the minimal model.

Variance of occupation number determined for all models.

Analytical solution available only for the minimal case.

Abstract

In random sequential adsorption (RSA), objects are deposited randomly, irreversibly, and sequentially; attempts leading to an overlap with previously deposited objects are discarded. The process continues until the system reaches a jammed state when no further additions are possible. We analyze a class of lattice RSA models in which landing on an empty site in a segment is allowed when at least $b$ neighboring sites on the left and the right are unoccupied. For the minimal model ($b=1$), we compute the full counting statistics of the occupation number. We reduce the determination of the full counting statistics to a Riccati equation that appears analytically solvable only when $b=1$. We develop a perturbation procedure which, in principle, allows one to determine cumulants consecutively, and we compute the variance of the occupation number for all $b$.