Abstract
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 Painlevé 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.
Mathematics Subject Classification (2000). Primary 47B35; Secondary 70H06.
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Krasovsky, I. (2011). Aspects of Toeplitz Determinants. In: Lenz, D., Sobieczky, F., Woess, W. (eds) Random Walks, Boundaries and Spectra. Progress in Probability, vol 64. Springer, Basel. https://doi.org/10.1007/978-3-0346-0244-0_16
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