Concentration inequalities for a removal-driven thinning process

We prove exponential concentration estimates and a strong law of large numbers for a particle system that is the simplest representative of a general class of models for 2D grain boundary coarsening introduced by the first author (2014). The system consists of $n$ particles in $(0,\infty )$ that move at unit speed to the left. Each time a particle hits the boundary point $0$, it is removed from the system along with a second particle chosen uniformly from the particles in $(0,\infty )$. Under the assumption that the initial empirical measure of the particle system converges weakly to a measure with density $f_0(x) \in L^1_+(0,\infty )$, the empirical measure of the particle system at time $t$ is shown to converge to the measure with density $f(x,t)$, where $f$ is the unique solution to the kinetic equation with nonlinear boundary coupling \begin{equation*} \partial _t f (x,t) - \partial _x f(x,t) = -\frac {f(0,t)}{\int _0^\infty f(y,t) dy} f(x,t), \quad 0<x < \infty , \end{equation*} and initial condition $f(x,0)=f_0(x)$.

The proof relies on a concentration inequality for an urn model studied by Pittel, and Maurey’s concentration inequality for Lipschitz functions on the permutation group.