2019/09/04 by Erol A. Peköz, Peköz, Erol, Adrian Röllin +3
Mathematics · Physics and Astronomy · #Stochastic processes and statistical mechanics #Random Matrices and Applications #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1909.01617
We study occupancy counts for the critical nearest-neighbor branching random\nwalk on the d-dimensional lattice, conditioned on non-extinction. For d\≥\n3, Lalley and Zheng (2011) showed that the properly scaled joint distribution\nof the number of sites occupied by j generation-n particles,\nj=1,2,\…, converges in distribution as n goes to infinity, to a\ndeterministic multiple of a single exponential random variable. The limiting\nexponential variable can be understood as the classical Yaglom limit of the\ntotal population size of generation n. Here we study the second order\nfluctuations around this limit, first, by providing a rate of convergence in\nthe Wasserstein metric that holds for all d\≥3, and second, by showing that\nfor d\≥ 7, the weak limit of the scaled joint differences between the\nnumber of occupancy-j sites and appropriate multiples of the total population\nsize converge in the Wasserstein metric to a multivariate symmetric Laplace\ndistribution. We also provide a rate of convergence for this latter result.\n