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Characteristic polynomials of random matrices. (English) Zbl 1042.82017
As stated by the authors, the correlation function of the eigenvalues of large \(N\times N\) matrices are known to exhibit a number of universal features in the large-\(N\) limit. In the Dyson limit, when the distances between these eigenvalues, measured in units of the local spacing, becomes of order \(1/N\), the correlation functions, as well as the level spacing distribution, become universal (independent of the specific probability measure). For finite differences, upon a smoothing of the distribution, the two-point correlation function is again universal, and the short-distance universality has been also shown to extend to external source problems, in which an external matrix is coupled to the random matrix.
The authors study the average of the characteristic polynomials, whose zeros are the eigenvalues of the random matrix. The probability distribution of the characteristic polynomial of a random matrix may be characterized by its moments, or better by its correlation functions. The study is motivated by various conjectures which have appeared recently in number theory for the zeros of the Riemann \(\zeta\)-function and its generalizations, the \(L\)-functions. The characteristic polynomials, as well as the zeta-fuctions, have their zeros on a straight line, and they obey the same statistical distribution.
Different number theorists have studied extensively the connections between the distribution of zeros of the Riemann \(\zeta\)-function, and of some generalizations thereof, with the statistics of the eigenvalues of large random matrices. The authors concentrate in particular in the comparison of the average moments of these functions in an interval to their counterpart in random matrices, which are the expectation values of the characteristic polynomials of the matrix.
It turns out that these expectation values have quite interesting interpretations. For instance, the moments of order \(2K\) scale, for unitary invariant ensembles, as the density of eigenvalues raised to the power \(K^2\). The prefactor turns out to be a universal number, i.e. it is independent of the specific probability distribution. An equivalent behaviour and prefactor had been found, as a conjecture, within number theory. The moments of the characteristic determinants of random matrices are computed here as limits, at coinciding points, of multi-point correlators of determinants. These correlators are in fact universal in Dyson’s scaling limit in which the difference between the points goes to zero, the size of the matrix goes to infinity, and their product remains finite.

MSC:
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
11Z05 Miscellaneous applications of number theory
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