Thek-point random matrix kernels obtained from one-point supermatrix models

Abstract

The k-point correlation functions of the Gaussian Random Matrix Ensembles are certain determinants of functions which depend on only two arguments. They are referred to as kernels, since they are the building blocks of all correlations. We show that the kernels are obtained, for arbitrary level number, directly from supermatrix models for one-point functions. More precisely, the generating functions of the one-point functions are equivalent to the kernels. This is surprising, because it implies that already the one-point generating function holds essential information about the k-point correlations. This also establishes a link to the averaged ratios of spectral determinants, i.e. of characteristic polynomials.

A new approach to derive Pfaffian structures for random matrix ensembles

“…The constant γ equals one for the real case and two for the quaternionic case. Such averages were considered before [34,24,18,35]. Here, we apply our method to show that the Pfaffian structure arises in a purely algebraic way.…”

“…4.1, we notice that the Pfaffian structure arising in all those ensembles is fundamental. Particularly, on the formal level, the obvious difference between matrix ensemble probability density P matrices in the probability probability for the matrices characteristic densities g(z 1 , z 2 ) density h(z) polynomials andg(z 1 , z 2 ) real symmetric matri-P (tr H m , m ∈ N) H P (x 1 )P (x 2 )× P (x)δ(y) ces [34,24,18] H = H T = H * ×δ(y 1 )δ(y 2 )Θ(x 2 − x 1 ) circular orthogonal P (tr U m , m ∈ N) U and U † P (e ıϕ 1 )P (e ıϕ 2 )× P (e ıϕ )δ(r − 1) ensemble [4] U † U = 1 1 N and 15,40,16,41,25] τ > 0 +2ıδ (2) (z 1 − z * 2 )Θ(y 1 )] [42,43,23] exp Table 1. Particular cases of the probability densities g(z 1 , z 2 ) and h(z) and their corresponding matrix ensembles of orthogonal rotation symmetry.…”

Giambelli compatible point processes

“…(both determinants have size d × d). By now this result has a number of different proofs and extensions, see [29], [57], [19], [20], [21], [22], [3], [18], [1], [4], [24], [31]. Formulas of this type are of interest in quantum physics and classical number theory, see [2], [28], [23], [33], [38], [39], [40].…”

Averages of characteristic polynomials in random matrix theory

“…However, the problem of computing the bulk scaling limit asymptotics of general averages (1.1.2), despite considerable interest of physicists, see e.g. Andreev-Simon [4], Gronqvist, Guhr and Kohler [37], Fyodorov [31], Szabo [54], Splttorff-Verbaarschot [52], Zirnbauer [59,60], remained open.…”