Totally positive Toeplitz matrices: classical and modern

By a theorem of Edrei, an infinite, normalised totally nonnegative upper-triangular Toeplitz matrix is determined by a pair of nonnegative parameter sequences, the `Schoenberg parameters', where nonzero parameters correspond to the roots and poles of a naturally associated generating function. These totally nonnegative Toeplitz matrices and their parameters also arise in the classification of characters of the infinite symmetric group by later work of Thoma. Moreover the Schoenberg parameters have an asymptotic interpretation in terms of irreducible representations of S_n and their Young diagrams by Vershik-Kerov. In this article we consider infinite totally positive Toeplitz matrices as limits of finite ones, and we obtain two further asymptotic descriptions of the Schoenberg parameters that are now related to quantum cohomology of the flag variety as n goes to infinity. One is related to asymptotics of normalised quantum parameters, and the other to asymptotics of the Chern classes of the tautological line bundles. We also describe the asymptotics of (quantum) Schubert classes in terms of the Schoenberg parameters. Our limit formulas relate to and were motivated by a tropical analogue of this theory that we survey. In the tropical setting one finds an asymptotic relationship between the `tropical Schoenberg parameters' and the weight map from Lusztig's parametrisation of the canonical basis.