Spectral relations between products and powers of isotropic random matrices

Abstract. In the past few years, the supersymmetry method was generalized to real-symmetric, Hermitean, and Hermitean self-dual random matrices drawn from ensembles invariant under the orthogonal, unitary, and unitary symplectic group, respectively. We extend this supersymmetry approach to chiral random matrix theory invariant under the three chiral unitary groups in a unifying way. Thereby we generalize a projection formula providing a direct link and, hence, a 'short cut' between the probability density in ordinary space and the one in superspace. We emphasize that this point was one of the main problems and critiques of the supersymmetry method since only implicit dualities between ordinary and superspace were known before. As examples we apply this approach to the calculation of the supersymmetric analogue of a Lorentzian (Cauchy) ensemble and an ensemble with a quartic potential. Moreover we consider the partially quenched partition function of the three chiral Gaussian ensembles corresponding to four-dimensional continuum QCD. We identify a natural splitting of the chiral Lagrangian in its lowest order into a part of the physical mesons and a part associated to source terms generating the observables, e.g. the level density of the Dirac operator.

Section: Discussionmentioning

confidence: 97%

“…This Gaussian models some kind of Dirac δ-function, i.e. we can "simplify" 35) where t/2 is the variance of the Gaussian distribution. Assuming that the integral of P multiplied with exp[|str σ 2 |] exists, we are allowed to interchange the integrations over Σ, V , and σ aux .…”

Section: Projection Formula For Dyson's Threefold Waymentioning

confidence: 99%

The supersymmetry method for chiral random matrix theory with arbitrary rotation-invariant weights

Kaymak¹,

Kieburg²,

Guhr³

2014

“…Haar matrices of the size (N + L) × (N + L), in which L columns and rows are removed. In the limit where both L and N tend to infinity in such a way that κ = L/N is fixed, F Y = κ r 2/n 1−r 2/n for r < (1 + κ) −n/2 and 1 otherwise [30], hence…”

Section: Examplesmentioning

confidence: 99%

Squared eigenvalue condition numbers and eigenvector correlations from the single ring theorem

Belinschi¹,

Nowak²,

Speicher³

et al. 2017

We extend the so-called "single ring theorem" [1], also known as the Haagerup-Larsen theorem [2], by showing that in the limit when the size of the matrix goes to infinity a particular correlator between left and right eigenvectors of the relevant non-hermitian matrix X, being the spectral density weighted by the squared eigenvalue condition number, is given by a simple formula involving only the radial spectral cumulative distribution function of X. We show that this object allows to calculate the conditional expectation of the squared eigenvalue condition number. We give examples and we provide cross-check of the analytic prediction by the large scale numerics.

“…[13,36] for M = 1 and [37] for general M . We see that the support of this function is a disk of radius µ M/2 which is smaller than the radius √ µ of a single matrix (M = 1) in the product since µ ≤ 1.…”

Section: Macroscopic Regimementioning

confidence: 99%

“…While studying the asymptotic behavior of the kernel in the vicinity of the origin for N → ∞ it is convenient to introduce a rescaled variable δz with |δz| of order one, 37) and express results in δz. One could alternatively use a variable δz ′ in the scaling formula z = N −M/2 δz ′ but since α = L/N is kept fixed in the limit N → ∞, δz and δz ′ differ by an inessential constant δz = α M/2 δz ′ which does not affect the N -dependence of the scaling.…”

Section: Fluctuations At the Originmentioning

confidence: 99%

Universal microscopic correlation functions for products of truncated unitary matrices

Akemann

Burda

Kieburg

et al. 2014

We investigate the spectral properties of the product of M complex non-Hermitian random matrices that are obtained by removing L rows and columns of larger unitary random matrices uniformly distributed on the group U (N + L). Such matrices are called truncated unitary matrices or random contractions. We first derive the joint probability distribution for the complex eigenvalues of the product matrix for fixed N, L, and M , given by a standard determinantal point process in the complex plane. The weight however is non-standard and can be expressed in terms of the Meijer G-function. The explicit knowledge of all eigenvalue correlation functions and the corresponding kernel allows us to take various large N (and L) limits at fixed M . At strong non-unitarity, with L/N finite, the eigenvalues condense on a domain inside the unit circle. At the edge and in the bulk we find the same universal microscopic kernel as for a single complex non-Hermitian matrix from the Ginibre ensemble. At the origin we find the same new universality classes labelled by M as for the product of M matrices from the Ginibre ensemble. Keeping a fixed size of truncation, L, when N goes to infinity leads to weak non-unitarity, with most eigenvalues on the unit circle as for unitary matrices. Here we find a new microscopic edge kernel that generalizes the known results for M = 1. We briefly comment on the case when each product matrix results from a truncation of different size L j .