Concentration inequalities for a removal-driven thinning process

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 introduced by the first author (2014). The system consists of n n particles in ( 0 , ∞ ) (0,\infty ) that move at unit speed to the left. Each time a particle hits the boundary point 0 0 , it is removed from the system along with a second particle chosen uniformly from the particles in ( 0 , ∞ ) (0,\infty ) . Under the assumption that the initial empirical measure of the particle system converges weakly to a measure with density f 0 ( x ) ∈ L + 1 ( 0 , ∞ ) f_0(x) \in L^1_+(0,\infty ) , the empirical measure of the particle system at time t t is shown to converge to the measure with density f ( x , t ) f(x,t) , where f f is the unique solution to the kinetic equation with nonlinear boundary coupling ∂ t f ( x , t ) − ∂ x f ( x , t ) = − f ( 0 , t ) ∫ 0 ∞ f ( y , t ) d y f ( x , t ) , 0 > x > ∞ , \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 ) 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.