Extreme values of CUE characteristic polynomials: a numerical study

Abstract

We present the results of systematic numerical computations relating to the extreme value statistics of the characteristic polynomials of random unitary matrices drawn from the Circular Unitary Ensemble (CUE) of Random Matrix Theory. In particular, we investigate a range of recent conjectures and theoretical results inspired by analogies with the theory of logarithmicallycorrelated Gaussian random fields. These include phenomena related to the conjectured freezing transition. Our numerical results are consistent with, and therefore support, the previous conjectures and theory. We also go beyond previous investigations in several directions: we provide the first quantitative evidence in support of a correlation between extreme values of the characteristic polynomials and large gaps in the spectrum, we investigate the rate of convergence to the limiting formulae previously considered, and we extend the previous analysis of the CUE to the CβE which corresponds to allowing the degree of the eigenvalue repulsion to become a parameter.arXiv:1806.00286v2 [cond-mat.stat-mech]

On the Moments of the Moments of the Characteristic Polynomials of Random Unitary Matrices

“…We note that the conjecture described above extends to the other circular ensembles (i.e. to the CβE) [9,21,33] and to the Gaussian ensembles [24][25][26]. We note as well that there are extensive mathematics and physics literatures on log-correlated Gaussian fields; see, for example [17], [26] and [8], and references contained therein.…”

“…Finally, our formulae have already been applied to analysing the results of numerical computations using randomly generated unitary matrices, where they explain the fluctuations in the moments of the characteristic polynomials evaluated by averaging over the unit circle [21]. We anticipate further similar applications and extensions to other numerical computations of the moments of spectral determinants.…”

“…The formulae we record extend the results of preliminary calculations due to Keating and Scott [32] (c.f. [29]), which formed the basis for some of the numerical computations in [21].…”

“…This observation, together with heuristic calculations and numerical experiments (c.f. [21]), motivated a series of conjectures [22,23] concerning the maximum of |P N (A, θ)| on the unit circle, (3) P max (A) = max 0≤θ<2π |P N (A, θ)|.…”

Moments of Moments and Branching Random Walks

“…In recent years there has been significant progress towards understanding the value distribution of the maximum of the logarithm of the characteristic polynomial of a random unitary matrix and of related log-correlated processes [1][2][3][4][5][6]11,[17][18][19][20][21][22][24][25][26][27]29]. Let the maximum value of P N (A, θ) around the unit circle.…”