Analysis of Large Urn Models with Local Mean-Field Interactions
Résumé
The stochastic models investigated in this paper describe the evolution of a set of $F_N$ identical balls scattered into $N$ urns connected by an underlying symmetrical graph with constant degree $h_N$. After some exponentially distributed amount of time {\em all the balls} of one of the urns are redistributed locally, among the $h_N$ urns of its neighborhood. The allocation of balls is done at random according to a set of weights which depend on the state of the system. The main original features of this context is that the cardinality $h_N$ of the range of interaction is not necessarily linear with respect to $N$ as in a classical mean-field context and, also, that the number of simultaneous jumps of the process is not bounded due to the redistribution of all balls of an urn at the same time. The approach relies on the analysis of the evolution of the local empirical distributions associated to the state of urns in the neighborhood of a given urn. Under some convenient conditions, by taking an appropriate Wasserstein distance and by establishing appropriate technical estimates for local empirical distributions, we are able to establish a mean-field convergence result. Convergence results of the corresponding invariant distributions are obtained for several allocation policies. For the class of power of $d$ choices policies for the allocations of balls, we show that the invariant measure of the corresponding McKean-Vlasov process has an asymptotic finite support property when the average load per urn gets large. This result differs somewhat from the classical double exponential decay property usually encountered in the literature for power of $d$ choices policies. This finite support property has interesting consequences in practice.