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The characteristic polynomial of a random unitary matrix and Gaussian multiplicative chaos.

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We consider the absolute value of the characteristic polynomial of a Haar distributed random unitary matrix. More precisely, we divide the absolute value of the characteristic polynomial by its expectation and think of this as the density of a random measure on the unit circle. Using some recent results on asymptotics of Toeplitz determinants by Tom Claeys and Igor Krasovsky, we prove that as one increases the size of the matrix, this measure converges to a random measure which can be described with the theory of Gaussian multiplicative chaos.

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