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{\bf Ioana Dumitriu and Etienne Rassart}
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{\bf Path Counting and Random Matrix Theory}
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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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