We propose a method for computing any Gelfand-Dickey τ function living in SegalWilson Grassmannian as the asymptotics of block Toeplitz determinant associated to a certain class of symbols W(t; z). Also truncated block Toeplitz determinants associated to the same symbols are shown to be τ function for rational reductions of KP. Connection with Riemann-Hilbert problems is investigated both from ...

Fisher-Hartwig asymptotics refers to the large n form of a class of Toeplitz determinants with singular generating functions. This class of Toeplitz determinants occurs in the study of the spin-spin correlations for the two-dimensional Ising model, and the ground state density matrix of the impenetrable Bose gas, amongst other problems in mathematical physics. We give a new application of the o...

In this paper we study matrices A = (aij) whose (i, j) -entry is a function of i/j; that is, aij = f(i/j) for some f : Q → C. We obtain a formula for the truncated determinants in the case where f is multiplicative, linking them to determinants of truncated Toeplitz matrices. We apply our formula to obtain several determinants of number-theoretic matrices. Mathematics Subject Classification 200...

Toeplitz operators (or equivalently, Wiener-Hopf operators; more generally, block Toeplitz operators; and particularly, Toeplitz determinants) are of importance in connection with a variety of problems in physics, and in particular, in the field of quantum mechanics. For example, a study of solvable models in quantum mechanics uses the spectral theory of Toeplitz operators (cf. [Pr]); the one-d...

By using Riemann–Hilbert problem based techniques, we obtain the asymptotic expansion of lacunary Toeplitz determinants detN [ cla−mb [ f ] ] generated by holomorhpic symbols, where la = a (resp. mb = b) except for a finite subset of indices a = h1, . . . , hn (resp. b = t1, . . . , tr). In addition to the usual Szegö asymptotics, our answer involves a determinant of size n + r.

We evaluate the Geiss-Leclerc-Schröer φ-map for shape modules over the preprojective algebra Λ of type c A1 in terms of matrix minors arising from the block-Toeplitz representation of the loop group SL2(L). Conjecturally these minors are among the cluster variables for coordinate rings of unipotent cells within SL2(L). In so doing we compute the Euler characteristic of any generalized flag vari...

We study the limiting behavior of TrUk(n), where U is a n × n random unitary matrix and k(n) is natural number that may vary with n in an arbitrary way. Our analysis is based on the connection with Toeplitz determinants. The central observation of this paper is a strong Szegö limit theorem for Toeplitz determinants associated to symbols depending on n in a particular way. As a consequence to th...

The asymptotic expansions of Toeplitz determinants of certain symbols with multiple jump discontinuities are shown to satisfy a revised version of the conjecture of Fisher and Hartwig. The Toeplitz matrix T n [φ] is said to be generated by the function φ if T n [φ] = (φ i−j), i, j = 0,. .. , n − 1. where φ n = 1 2π 2π 0 φ(θ)e −inθ dθ is the nth Fourier coefficient of φ. Define the determinant T...