Dumitriu, Ioana; Rassart, Etienne Path counting and random matrix theory. (English) Zbl 1031.05017 Electron. J. Comb. 10, Research paper R43, 16 p. (2003); printed version J. Comb. 10, No. 4 (2003). Summary: 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. Cited in 3 Documents MSC: 05A19 Combinatorial identities, bijective combinatorics 15B52 Random matrices (algebraic aspects) 82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics Keywords:identities; Dyck paths; Motzkin paths; random walks; random matrix PDFBibTeX XMLCite \textit{I. Dumitriu} and \textit{E. Rassart}, Electron. J. Comb. 10, No. 1, Research paper R43, 16 p. (2003; Zbl 1031.05017) Full Text: arXiv EuDML EMIS