T\'oth's buses and the "detachment process''

TL;DR

This paper introduces the detachment process, a Markov process inspired by Toth's problem on passenger seating, analyzing detachment times, clumping, and the effects of different time scales on passenger distribution.

Contribution

It extends Toth's model by treating the number of buses as a time parameter, creating a coupled process called the detachment process, and analyzes its properties and critical time scales.

Findings

Identification of four critical time scales governing the process

Analysis of detachment and clumping behaviors

Introduction of a comparison theorem for binomial distributions

Abstract

This paper introduces the \textbf{detachment process}, a novel, time-inhomogeneous Markov process inspired by I. P. T\'oth's problem \cite{Toth} concerning the number of ``lonely passengers'' (those without companions) when $n$ passengers are seated independently and uniformly in $k$ initially empty buses. T\'oth showed that this number is stochastically non-decreasing in $k$ for fixed $n$ (see also Haslegrave's work \cite{Haslegrave}). We extend T\'oth's model by treating the number of buses $k$ as a time parameter. Specifically, for a fixed number of passengers $n$, the state of our Markov process at time $k \ge 1$ is exactly T\'oth's configuration $(n, k)$. (We formally extend the process definition for all $t \in [1, \infty)$.) These processes can be coupled for all $n \ge 1$, and this larger coupled process is what we dub the \textbf{detachment process}. Our investigation focuses on properties related to detachment, clumping, the number of lonely passengers and of non-empty buses. The central notion is \textbf{detachment}, which occurs at time $k$ if every passenger occupies a distinct bus; we say the process is \textbf{in a state of detachment} at $k$. A \textbf{detachment time} $k$ is when the process transitions from a non-detached state at $k-1$ to a detached state at $k$. \textbf{Four critical time scales} are idetified -- linear, quadratic, and log-corrected linear or quadratic in the number of passengers, $n$ -- that govern the process's properties. We investigate (relative) clumping. We also explore why modeling the number of passengers with a Poisson distribution simplifies the analysis of T\'oth's original model. To aid this derivation, we introduce a comparison theorem for binomial distributions, originally obtained by J. Najnudel \cite{Najnudel}, along with a novel proof.