Asymptotics for the Number of Random Walks in the Euclidean Lattice

Autor:	Dumitraşcu, Dorin, Suciu, Liviu
Rok vydání:	2022
Předmět:	Mathematics - Combinatorics Mathematics - Probability 05A16 (Primary) 05A10 40E99 33C45 (Secondary)
Druh dokumentu:	Working Paper
Popis:	We give precise asymptotics to the number of random walks in the standard orthogonal lattice in $\mathbb{R}^d$ that return to the starting point at step $2n$, both for all such walks and for the ones that return for the first time. The first set of asymptotics is obtained in an elementary way, by using a combinatorial and geometric multiplication principle together with the classical theory of Legendre polynomials. As an easy consequence we obtain a unified proof of P\'{o}lya's theorem. By showing that the relevant generating functions are $\Delta$-analytic, we use the deeper Tauberian theory of singularity analysis to obtain the asymptotics for the first return paths. Comment: 26 pages, 6 figures
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2212.01702 Zobrazit plný text záznamu View this record from Arxiv