The asymptotics of monotone subsequences of involutions

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  • 23 March 2000
Abstract
We consider the asymptotics of the distributions of the lengths of the longest monotone subsequences of random involutions taken from various ensembles. They are ensembles of involutions, signed involutions, involutions with constraint on the number of fixed points and signed involutions with constraint on the number of fixed and negated points. The resulting limiting distributions are expressed in terms of either the Tracy-Widom distributions for the largest eigenvalues of random GOE, GUE, GSE matrices, or new classes of distributions defined in terms of the solution of the Riemann-Hilbert problem for the Painlev\'e II equation. These new classes of distributions interpolate between certain pairs of the Tracy-Widom distributions. We also consider the asymptotic distributions of the length of the second rows of the corresponding Young diagram ensembles. In each case, convergence of moments is also obtained.

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