Growth-fragmentation processes describe the evolution of systems in which cells grow slowly and fragment suddenly. Despite originating as a way to describe biological phenomena, they have recently been found to describe the lengths of certain curves in statistical physics models. In this note, we describe a new growth-fragmentation process connected to random planar maps with faces of large degree, having as a key ingredient the ricocheted stable process recently discovered by Budd. The process has applications to the excursions of planar Brownian motion and Liouville quantum gravity.

Let ${{R}_{n}}\left( \alpha \right)$ be the $n!\,\times \,n!$ matrix whose matrix elements ${{\left[ {{R}_{n}}\left( \alpha \right) \right]}_{\sigma p}}$ , with $\sigma$ and $p$ in the symmetric group ${{G}_{n}}$ , are ${{\alpha }^{\ell \left( \sigma {{p}^{-1}} \right)}}$ with $0\,<\,\alpha \,<\,1$, where $\ell \left( \text{ }\!\!\pi\!\!\text{ } \right)$ denotes the number of cycles in $\text{ }\pi \text{ }\in {{G}_{n}}.$ We give the spectrum of ${{R}_{n}}$ and show that the ratio of the largest eigenvalue ${{\text{ }\!\!\lambda\!\!\text{ }}_{0}}$ to the second largest one (in absolute value) increases as a positive power of $n$ as $n\,\to \,\infty$.

The fluctuation of the counts of particles in m regions are used to yield an estimate of a fundamental motion parameter. The effect of varying m on the precision of the estimation method is considered. An exact analysis and simulation of the Fürth pedestrian vector process is presented.

This article deals with the generalized Smoluchowski process, {n(t), t ≧ 0}, defined by the temporal fluctuating of the numbers of randomly moving particles contained in m < ∞ disjoint regions of space. The relationship of the Smoluchowski process {n(t), t ≧ 0} to the emigration–immigration process is discussed and conditions for their equivalence are presented.