The asymptotics of monotone subsequences of involutions

We compute the limiting distributions of the lengths of the longest monoton e subsequences of random (signed) involutions with or without conditions on the number of fixed points (and negated points) as the sizes of the involu tions tend to infinity. The resulting distributions are, depending on the n umber of fixed points, (1) the Tracy-Widom distributions for the largest ei genvalues of random GOE, GUE, GSE matrices, (2) the normal distribution, or (3) new classes of distributions which interpolate between pairs of the Tr acy-Widom distributions. We also consider the second rows of the correspond ing Young diagrams. In each case the convergence of moments is also shown. The proof is based on the algebraic work of J. Baik and E. Rains in [7] whi ch establishes a connection between the statistics of random involutions an d a family of orthogonal polynomials, and an asymptotic analysis of the ort hogonal polynomials which is obtained by extending the Riemann-Hilbert anal ysis for the orthogonal polynomials by P Deift, K. Johansson, and Baik in [ 3].