Tóth's buses and the "detachment process''

This paper introduces the $\mathbf{\text{detachment}}\mathbf{\text{process}}$, a novel, time-inhomogeneous Markov process inspired by I. P. Tóth’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. Tóth showed that this number is stochastically non-decreasing in $k$ for fixed $n$ (see also Haslegrave’s work \cite{Haslegrave}). We extend Tóth’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 Tóth’s configuration $(n,k)$. (We formally extend the process definition for all $t\in [1,\mathrm{\infty })$.) These processes can be coupled for all $n\ge 1$, and this larger coupled process is what we dub the $\mathbf{\text{detachment}}\mathbf{\text{process}}$. Our investigation focuses on properties related to detachment, clumping, the number of lonely passengers and of non-empty buses. The central notion is $\mathbf{\text{detachment}}$, which occurs at time $k$ if every passenger occupies a distinct bus; we say the process is $\mathbf{\text{in}}\mathbf{\text{a}}\mathbf{\text{state}}\mathbf{\text{of}}\mathbf{\text{detachment}}$ at $k$. A $\mathbf{\text{detachment}}\mathbf{\text{time}}$ $k$ is when the process transitions from a non-detached state at $k-1$ to a detached state at $k$. $\mathbf{\text{Four}}\mathbf{\text{critical}}\mathbf{\text{time}}\mathbf{\text{scales}}$ 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 Tóth’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.