# King's College London

## On the distribution of the maximum value of the characteristic polynomial of GUE random matrices

Research output: Contribution to journalArticlepeer-review

### Standard

On the distribution of the maximum value of the characteristic polynomial of GUE random matrices. / Fyodorov, Yan V; J Simm, N.

In: NONLINEARITY, Vol. 29, No. 9, 10.08.2016, p. 2837–2855.

Research output: Contribution to journalArticlepeer-review

### Harvard

Fyodorov, YV & J Simm, N 2016, 'On the distribution of the maximum value of the characteristic polynomial of GUE random matrices', NONLINEARITY, vol. 29, no. 9, pp. 2837–2855. https://doi.org/10.1088/0951-7715/29/9/2837

### APA

Fyodorov, Y. V., & J Simm, N. (2016). On the distribution of the maximum value of the characteristic polynomial of GUE random matrices. NONLINEARITY, 29(9), 2837–2855. https://doi.org/10.1088/0951-7715/29/9/2837

### Vancouver

Fyodorov YV, J Simm N. On the distribution of the maximum value of the characteristic polynomial of GUE random matrices. NONLINEARITY. 2016 Aug 10;29(9): 2837–2855. https://doi.org/10.1088/0951-7715/29/9/2837

### Author

Fyodorov, Yan V ; J Simm, N. / On the distribution of the maximum value of the characteristic polynomial of GUE random matrices. In: NONLINEARITY. 2016 ; Vol. 29, No. 9. pp. 2837–2855.

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title = "On the distribution of the maximum value of the characteristic polynomial of GUE random matrices",
abstract = "Motivated by recently discovered relations between logarithmically correlated Gaussian processes and characteristic polynomials of large random $N\times N$ matrices H from the Gaussian unitary ensemble (GUE), we consider the problem of characterising the distribution of the global maximum of ${{D}_{N}}(x):=\log |\det (xI-H)|$ as $N\to \infty$ and $x\in (-1,1)$ . We arrive at an explicit expression for the asymptotic probability density of the (appropriately shifted) maximum by combining the rigorous Fisher–Hartwig asymptotics due to Krasovsky [34] with the heuristic freezing transition scenario for logarithmically correlated processes. Although the general idea behind the method is the same as for the earlier considered case of the circular unitary ensemble, the present GUE case poses new challenges. In particular we show how the conjectured self-duality in the freezing scenario plays the crucial role in our selection of the form of the maximum distribution. Finally, we demonstrate a good agreement of the found probability density with the results of direct numerical simulations of the maxima of D N  (x).",
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AU - Fyodorov, Yan V

AU - J Simm, N

PY - 2016/8/10

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N2 - Motivated by recently discovered relations between logarithmically correlated Gaussian processes and characteristic polynomials of large random $N\times N$ matrices H from the Gaussian unitary ensemble (GUE), we consider the problem of characterising the distribution of the global maximum of ${{D}_{N}}(x):=\log |\det (xI-H)|$ as $N\to \infty$ and $x\in (-1,1)$ . We arrive at an explicit expression for the asymptotic probability density of the (appropriately shifted) maximum by combining the rigorous Fisher–Hartwig asymptotics due to Krasovsky [34] with the heuristic freezing transition scenario for logarithmically correlated processes. Although the general idea behind the method is the same as for the earlier considered case of the circular unitary ensemble, the present GUE case poses new challenges. In particular we show how the conjectured self-duality in the freezing scenario plays the crucial role in our selection of the form of the maximum distribution. Finally, we demonstrate a good agreement of the found probability density with the results of direct numerical simulations of the maxima of D N  (x).

AB - Motivated by recently discovered relations between logarithmically correlated Gaussian processes and characteristic polynomials of large random $N\times N$ matrices H from the Gaussian unitary ensemble (GUE), we consider the problem of characterising the distribution of the global maximum of ${{D}_{N}}(x):=\log |\det (xI-H)|$ as $N\to \infty$ and $x\in (-1,1)$ . We arrive at an explicit expression for the asymptotic probability density of the (appropriately shifted) maximum by combining the rigorous Fisher–Hartwig asymptotics due to Krasovsky [34] with the heuristic freezing transition scenario for logarithmically correlated processes. Although the general idea behind the method is the same as for the earlier considered case of the circular unitary ensemble, the present GUE case poses new challenges. In particular we show how the conjectured self-duality in the freezing scenario plays the crucial role in our selection of the form of the maximum distribution. Finally, we demonstrate a good agreement of the found probability density with the results of direct numerical simulations of the maxima of D N  (x).

U2 - 10.1088/0951-7715/29/9/2837

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EP - 2855

JO - NONLINEARITY

JF - NONLINEARITY

SN - 0951-7715

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