We summarize some of the recent developments which link certain problems in combinatorial theory related to random growth to random matrix theory. 2000 Mathematics Subject Classification: 60C05.

as the Fredholm determinant of an operator 1−K acting on l2({n, n+1, . . . }), where the kernel K = K(φ) admits an integral representation in terms of φ. The answer is affirmative and the construction of the kernel is explained below. We give two versions of the result: an algebraic one, which holds in the suitable algebra of formal power series, and an analytic one. In order to minimize the am...

The paper is devoted to exact and asymptotic formulas for the determinants of Toeplitz matrices with perturbations by blocks of fixed size in the four corners. If the norms of the inverses of the unperturbed matrices remain bounded as the matrix dimension goes to infinity, then standard perturbation theory yields asymptotic expressions for the perturbed determinants. This premise is not satisfi...

In this paper we study the asymptotic behavior, as n → ∞, of ratios Dn(e hdμ)/Dn(dμ) of Toeplitz determinants defined by a measure μ and a sufficiently smooth function h. The approach we follow is based on the Verblunsky coefficients associated to μ. We prove that that the second order asymptotics depends on only few Verblunsky coefficients. In particular, we establish a relative version of the...

We consider the Toeplitz matrices and obtain their unique LU factor-izations. As by-products, we get an explicit formula for the determinant of a Toeplitz matrix and the application of inversion of Toeplitz matrices .

We describe a simple one-person card game, patience sorting. Its analysis leads to a broad circle of ideas linking Young tableaux with the longest increasing subsequence of a random permutation via the Schensted correspondence. A recent highlight of this area is the work of Baik-Deift-Johansson which yields limiting probability laws via hard analysis of Toeplitz determinants.

The first Szegő limit theorem has been extended by Bump-Diaconis and Tracy-Widom to limits of other minors of Toeplitz matrices. We extend their results still further to allow more general measures and more general determinants. We also give a new extension to higher dimensions, which extends a theorem of Helson and Lowdenslager. §

We describe a simple one-person card game, patience sorting. Its analysis leads to a broad circle of ideas linking Young tableaux with the longest increasing subsequence of a random permutation via the Schensted correspondence. A recent highlight of this area is the work of Baik-Deift-Johansson which yields limiting probability laws via hard analysis of Toeplitz determinants.

The purpose of this article is to obtain some new infinite families of Toeplitz matrices, 7-matrices and generalized Pascal triangles whose leading principal minors form the Fibonacci, Lucas, Pell and Jacobsthal sequences. We also present a new proof for Theorem 3.1 in [R. Bacher. Determinants of matrices related to the Pascal triangle. J. Théor. Nombres Bordeaux, 14:19–41, 2002.].

We give alternative proofs to (block case versions of) some formulas for Toeplitz and Fredholm determinants established recently by the authors of the title. Our proof of the Borodin-Okounkov formula is very short and direct. The proof of the Baik-Deift-Rains formulas is based on standard manipulations with Wiener-Hopf factorizations.