2022/12/03 by Dorin Dumitraşcu, Dumitraşcu, Dorin, Liviu Suciu +1 · 2 citations
Mathematics · #05A16 (Primary) 05A10 #33C45 (Secondary) #40E99 #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2212.01702
openalex publication_date 2022/12/03 · openalex created_date 2022/12/18 · openalex updated_date 2026/07/28
We give precise asymptotics to the number of first time returning random walks in the standard orthogonal lattice in ℝ and we prove that these numbers do not form a P-recursive sequence. In the process, the known asymptotics of the number of closed walks are obtained in an elementary way, by using a combinatorial and geometric multiplication principle together with the classical theory of Legendre polynomials. By showing that the relevant generating functions are G-functions, we use a form of the Hadamard convolution to find their singularities in all dimensions and give the ODEs that they satisfy for d≤ 5, some of which seem to be new. We use the Lucas property of the number of closed walks to prove that the corresponding generating function is not invertible as a G-function, which immediately implies that the generating function of the first time returning walks is not holonomic. We propose a few conjectures on the form of the asymptotic coefficients and of the ODEs.