Random Matrix Theory Ii: Algebraic Developments

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Encyclopedia of Nonlinear Science

It was hypothesized by Eugene Wigner in the 1950s that the highly excited states of complex nuclei would have the same statistical properties as the eigenvalues of a large random real symmetric matrix. In pure mathematics, one finds a random matrix hypothesis in the theory of the celebrated Riemann hypothesis. Thus the Montgomery-Odlyzko law states that the statistics of the large zeros of the Riemann zeta function on the critical line (Riemann zeros) coincide with the statistics of the eigenvalues of a large complex Hermitian matrix. To test such hypotheses (for definiteness, the Montgomery-Odlyzko law), one computes a large sequence of consecutive Riemann zeros, scales the sequence so that locally the mean spacing is unity, and then empirically computes statistical quantities. A typical example of the latter is the distribution of the spacing between consecutive zeros. This must be compared against the same statistical quantity for the eigenvalues of large random complex Hermitian matrices. How then does one compute the eigenvalue spacing distribution for random matrices?

Hermitian random matrices with real, complex, and quaternion real Gaussian elements form matrix ensembles referred to as the Gaussian orthogonal ensemble (GOE) (β=1), the Gaussian unitary ensemble (GUE) (β=2), and the Gaussian symplectic ensemble (GSE) (β=4), respectively, where β is a convenient label. Consider the bulk eigenvalues of such large matrices, scaled to have unit density. Denote by p_β(k; s) the probability density that there are exactly k eigenvalues in between two eigenvalues of spacing s, and denote by E_β(k; s) the probability that there are exactly k eigenvalues in an interval of size s. Define the generating functions and Note from the definitions that these quantities are related by

Gaudin observed that the determinantal form of the correlations in the case β=2 allows E₂(s; ξ) to be written as a Fredholm determinant,

where K₂, are integral operators on (0, s) with kernels

and

respectively. The second equality is noted in (1) because both factors therein are related to E₁. Thus, with E₁(−1; s)=0, define

Then an inter-relationship between large GOE and large GUE matrices due to Dyson implies

This factorization turns out to be the same as in (1), so one obtains Mehta’s result

For the case β=4, one uses Mehta’s and Dyson’s interrelationship between large GSE matrices and large GOE matrices to conclude

A new line of study of E_β was initiated by Jimbo, Miwa, Môri, and Sato in 1980, which related the Fredholm determinant in (1) to integrable systems theory, resulting in the formula

where σ satisfies a particular example of the σ-form of the Painlevé V equation,

The quantities can also be expressed in terms of Painlevé transcendents. Thus, combining results of the present author with results of Tracy and Widom, one has

where υ(t; ξ; a) satisfies a particular example of the σ-form of the Painlevé III′ equation

Another bulk spacing distribution with a Painlevé type evaluation is the nearest-neighbor spacing between

eigenvalues (i.e., the minimum of the distances to the left neighbor and the right neighbor) for large GUE matrices, nn(t) say. Forrester and Odlyzko (1996) have shown that

where y(s; a) satisfies the differential equation

subject to the boundary condition

Comparison of a plot of this statistic, obtained from the formula in Forrester and Odlyzko (1996), against the same statistic computed, empirically for large sequences of Riemann zeros starting at three different positions along the critical line is given in Figure 1.

P.J.FORRESTER

See also Random matrix theory I, III, IV

Forrester, P.J. Eigenvalue probabilities and Painlevé theory, Chapter 6 of Log-gases and random matrices, www.ms.unimelb.edu.au/~matpjf/matpjf.html

Forrester, P.J. & Odlyzko, A.M. 1996. Gaussian unitary ensemble eigenvalues and Riemann ζ function zeros: a non-linear equation for a new statistic, Physical Review E, 54:R4493– R4495

Tracy, C.A. & Widom, H. 1993. Introduction to random matrices. In Geometric and Quantum Aspects of Integrable Systems, edited by G.F.Helminck, New York: Springer, pp. 407–424

van Moerbeke, P. 2001. Integrable lattices: random matrices and random permutations. In Random Matrices and Their Applications, edited by P.Bleher & A.Its, Cambridge and New York: Cambridge University Press, pp. 321–406

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