A Distribution Function Arising in Computational Biology

Karlin and Altschul in their statistical analysis for multiple high-scoring segments in molecular sequences introduced a distribution function which gives the probability there are at least r distinct and consistently ordered segment pairs all with score at least x. For long sequences this distribution can be expressed in terms of the distribution of the length of the longest increasing subsequence in a random permutation. Within the past few years, this last quantity has been extensively studied in the mathematics literature. The purpose of these notes is to summarize these new mathematical developments in a form suitable for use in computational biology.Comment: 9 pages, no figures. Revised version makes minor change

Correlation Functions, Cluster Functions and Spacing Distributions for Random Matrices

The usual formulas for the correlation functions in orthogonal and symplectic matrix models express them as quaternion determinants. From this representation one can deduce formulas for spacing probabilities in terms of Fredholm determinants of matrix-valued kernels. The derivations of the various formulas are somewhat involved. In this article we present a direct approach which leads immediately to scalar kernels for unitary ensembles and matrix kernels for the orthogonal and symplectic ensembles, and the representations of the correlation functions, cluster functions and spacing distributions in terms of them.Comment: 22 pages. LaTeX file. Minor correctio