Increasing Subsequences and the Classical Groups

We show that the moments of the trace of a random unitary matrix have combinatorial interpretations in terms of longest increasing subsequences of permutations. To be precise, we show that the 2n-th moment of the trace of a random k-dimensional unitary matrix is equal to the number of permutations of length n with no increasing subsequence of length greater than k. We then generalize this to other expectations over the unitary group, as well as expectations over the orthogonal and symplectic groups. In each case, the expectations count objects with restricted "increasing subsequence" length

Algebraic aspects of increasing subsequences

We present a number of results relating partial Cauchy-Littlewood sums, integrals over the compact classical groups, and increasing subsequences of permutations. These include: integral formulae for the distribution of the longest increasing subsequence of a random involution with constrained number of fixed points; new formulae for partial Cauchy-Littlewood sums, as well as new proofs of old formulae; relations of these expressions to orthogonal polynomials on the unit circle; and explicit bases for invariant spaces of the classical groups, together with appropriate generalizations of the straightening algorithm.Comment: LaTeX+amsmath+eepic; 52 pages. Expanded introduction, new references, other minor change

Growth models, random matrices and Painleve transcendents

The Hammersley process relates to the statistical properties of the maximum length of all up/right paths connecting random points of a given density in the unit square from (0,0) to (1,1). This process can also be interpreted in terms of the height of the polynuclear growth model, or the length of the longest increasing subsequence in a random permutation. The cumulative distribution of the longest path length can be written in terms of an average over the unitary group. Versions of the Hammersley process in which the points are constrained to have certain symmetries of the square allow similar formulas. The derivation of these formulas is reviewed. Generalizing the original model to have point sources along two boundaries of the square, and appropriately scaling the parameters gives a model in the KPZ universality class. Following works of Baik and Rains, and Pr\"ahofer and Spohn, we review the calculation of the scaled cumulative distribution, in which a particular Painlev\'e II transcendent plays a prominent role.Comment: 27 pages, 5 figure

Increasing subsequences and the hard-to-soft edge transition in matrix ensembles

Our interest is in the cumulative probabilities Pr(L(t) \le l) for the maximum length of increasing subsequences in Poissonized ensembles of random permutations, random fixed point free involutions and reversed random fixed point free involutions. It is shown that these probabilities are equal to the hard edge gap probability for matrix ensembles with unitary, orthogonal and symplectic symmetry respectively. The gap probabilities can be written as a sum over correlations for certain determinantal point processes. From these expressions a proof can be given that the limiting form of Pr(L(t) \le l) in the three cases is equal to the soft edge gap probability for matrix ensembles with unitary, orthogonal and symplectic symmetry respectively, thereby reclaiming theorems due to Baik-Deift-Johansson and Baik-Rains.Comment: LaTeX, 19 page

Universality of the Distribution Functions of Random Matrix Theory. II

This paper is a brief review of recent developments in random matrix theory. Two aspects are emphasized: the underlying role of integrable systems and the occurrence of the distribution functions of random matrix theory in diverse areas of mathematics and physics.Comment: 17 pages, 3 figure

Limit points of subsequences

Let $x$ be a sequence taking values in a separable metric space and $\mathcal{I}$ be a generalized density ideal or an $F_\sigma$-ideal on the positive integers (in particular, $\mathcal{I}$ can be any Erd{\H o}s--Ulam ideal or any summable ideal). It is shown that the collection of subsequences of $x$ which preserve the set of $\mathcal{I}$-cluster points of $x$ [respectively, $\mathcal{I}$-limit points] is of second category if and only if the set of $\mathcal{I}$-cluster points of $x$ [resp., $\mathcal{I}$-limit points] coincides with the set of ordinary limit points of $x$; moreover, in this case, it is comeager. In particular, it follows that the collection of subsequences of $x$ which preserve the set of ordinary limit points of $x$ is comeager.Comment: To appear in Topology Appl. arXiv admin note: substantial text overlap with arXiv:1711.0426

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