Concentration inequalities for a removal-driven thinning process

TL;DR

This paper establishes exponential concentration bounds and a strong law of large numbers for a particle system modeling 2D grain boundary coarsening, proving convergence to a solution of a nonlinear kinetic equation.

Contribution

It introduces new concentration inequalities for a removal-driven thinning process and proves convergence to a kinetic PDE with nonlinear boundary conditions.

Findings

Empirical measure converges to the solution of the kinetic equation.

Established exponential concentration estimates for the particle system.

Proved a strong law of large numbers for the system's empirical measure.

Abstract

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. 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 $$\partial_t f (x,t) - \partial_x f(x,t) = -\frac{f(0,t)}{\int_0^\infty f(y,t)\, dy} f(x,t), \quad 0<x < \infty, $$ 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.