The asymptotics of reflectable weighted walks in arbitrary dimension

Abstract

We consider the weighted lattice walks with a reflectable step set restricted to the positive $d$-dimensional orthant. We obtain asymptotic formulas for the number of such walks as a function of the weights. To do so, we set up the desired generating function as the diagonal of a rational function. Then we perform a coefficient extraction via an integral computation which is broken up into two cases. One part uses the residue theorem to evaluate the integral within an error, while the other uses known approximations of Fourier-Laplace integrals.