Aspects of Toeplitz determinants

Details Aspects of Toeplitz determinants Aspects of Toeplitz determinants

2010-07-07

arxiv.org/abs/1007.1128v3

Progress in Probability 64 (2011), 305-324

Physics

Mathematical Physics

16 pages, added references; in "Boundaries and Spectra of Random Walks" (D.Lenz, F.Sobieczky, W.Woess, editors)

Scientific paper

We review the asymptotic behavior of a class of Toeplitz (as well as related Hankel and Toeplitz + Hankel) determinants which arise in integrable models and other contexts. We discuss Szego, Fisher-Hartwig asymptotics, and how a transition between them is related to the Painleve V equation. Certain Toeplitz and Hankel determinants reduce, in certain double-scaling limits, to Fredholm determinants which appear in the theory of group representations, in random matrices, random permutations and partitions. The connection to Toeplitz determinants helps to evaluate the asymptotics of related Fredholm determinants in situations of interest, and we review the corresponding results.

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