Path counting and random matrix theory

Details Path counting and random matrix theory Path counting and random matrix theory

2003-07-17

arxiv.org/abs/math/0307252v1

Mathematics

Combinatorics

14 pages, 13 figures and diagrams; submitted to the Electronic Journal of Combinatorics

Scientific paper

We establish three identities involving Dyck paths and alternating Motzkin paths, whose proofs are based on variants of the same bijection. We interpret these identities in terms of closed random walks on the halfline. We explain how these identities arise from combinatorial interpretations of certain properties of the $\beta$-Hermite and $\beta$-Laguerre ensembles of random matrix theory. We conclude by presenting two other identities obtained in the same way, for which finding combinatorial proofs is an open problem.

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