Averages of characteristic polynomials in random matrix theory

Abstract

We compute averages of products and ratios of characteristic polynomials associated with orthogonal, unitary, and symplectic ensembles of random matrix theory. The Pfaffian/determinantal formulae for these averages are obtained, and the bulk scaling asymptotic limits are found for ensembles with Gaussian weights. Classical results for the correlation functions of the random matrix ensembles and their bulk scaling limits are deduced from these formulae by a simple computation. We employ a discrete approximation method: the problem is solved for discrete analogues of random matrix ensembles originating from representation theory, and then a limit transition is performed. Exact Pfaffian/determinantal formulae for the discrete averages are proven using standard tools of linear algebra; no application of orthogonal or skew‐orthogonal polynomials is needed. © 2005 Wiley Periodicals, Inc.

Massive partition functions and complex eigenvalue correlations in matrix models with symplectic symmetry

“…(2.23) was derived in [21] for the Gaussian SE in terms of expectation values of two characteristic polynomials, or equivalently in terms of the I N (x, t)-kernel [26]. In taking the Hermitean limit we now understand why only the I-and not the S-and D-kernels appear.…”

“…In taking the Hermitean limit we now understand why only the I-and not the S-and D-kernels appear. Moreover, the result from [21] is strictly valid only for a Gaussian weight function, whereas we can allow for an arbitrary weight here (for details see the derivation below) 5 . For the chiral ensembles we recover the results of [24,25].…”

“…It is not clear how to extend the results for ratios in the unitary case [19,20] to symplectic ensembles. The proof for ratios in the SE with real eigenvalues [21] is based on a discretisation of the ensemble which is not obvious to extend to the complex case.…”

A comparison of the superbosonization formula and the generalized Hubbard–Stratonovich transformation

“…We use the complex representation of the quaternionic numbers H. Also, we define γ 1 = 1 , β ∈ {2, 4} 2 , β = 1 , γ 2 = 1 , β ∈ {1, 2} 2 , β = 4 (2.1) andγ = γ 1 γ 2 . The central objects in many applications of supersymmetry are averages over ratios of characteristic polynomials [23,24,25] …”

Applications of theL-functions ratios conjectures

“…The point is that these ratios conjectures are useful for calculating both local and global statistics. In fact, quoting from [4], 'The averages of products and ratios of characteristic polynomials are more fundamental characteristics of random matrix models than the correlation functions. ' We would argue that the same can be said for L-functions.…”