Correlation functions for random involutions

Abstract

Our interest is in the scaled joint distribution associated with $k$-increasing subsequences for random involutions with a prescribed number of fixed points. We proceed by specifying in terms of correlation functions the same distribution for a Poissonized model in which both the number of symbols in the involution, and the number of fixed points, are random variables. From this, a de-Poissonization argument yields the scaled correlations and distribution function for the random involutions. These are found to coincide with the same quantities known in random matrix theory from the study of ensembles interpolating between the orthogonal and symplectic universality classes at the soft edge, the interpolation being due to a rank 1 perturbation.

27 pages, 1 figure, minor corrections made

References in corpus (6)

• Level-Spacing Distributions and the Airy Kernel
• The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
• Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges
• Fluctuations of the one-dimensional polynuclear growth model with external sources
• Correlation functions for symmetrized increasing subsequences
• Polynuclear growth on a flat substrate and edge scaling of GOE eigenvalues