  We compute the limiting distributions of the lengths of the longest monotone
subsequences of random (signed) involutions with or without conditions on the
number of fixed points (and negated points) as the sizes of the involutions
tend to infinity. The resulting distributions are, depending on the number of
fixed points, (1) the Tracy-Widom distributions for the largest eigenvalues of
random GOE, GUE, GSE matrices, (2) the normal distribution, or (3) new classes
of distributions which interpolate between pairs of the Tracy-Widom
distributions. We also consider the second rows of the corresponding Young
diagrams. In each case the convergence of moments is also shown. The proof is
based on the algebraic work of the authors in \cite{PartI} which establishes a
connection between the statistics of random involutions and a family of
orthogonal polynomials, and an asymptotic analysis of the orthogonal
polynomials which is obtained by extending the Riemann-Hilbert analysis for the
orthogonal polynomials by Deift, Johansson and the first author in [BDJ].
