Abstract
In this article we describe how Toeplitz determinants arise in the theory of random matrices (the distribution of eigenvalues of a random unitary matrix) and in a problem concerning random permutations (the length of the longest increasing subsequence in a random permutation). In the first case the connection is direct and in the second case the connection is indirect, via some highly nontrivial combinatorics. In all examples the problem can be recast as one of determining the asymptotics of large Toeplitz determinants with variable symbol.
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Widom, H. (2002). Toeplitz Determinants, Random Matrices and Random Permutations. In: Böttcher, A., Gohberg, I., Junghanns, P. (eds) Toeplitz Matrices and Singular Integral Equations. Operator Theory: Advances and Applications, vol 135. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8199-9_19
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DOI: https://doi.org/10.1007/978-3-0348-8199-9_19
Publisher Name: Birkhäuser, Basel
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