• Corpus ID: 119601092

Howe pairs, supersymmetry, and ratios of random characteristic polynomials for the unitary groups U(N)

  title={Howe pairs, supersymmetry, and ratios of random characteristic polynomials for the unitary groups U(N)},
  author={J. Brian Conrey and David W. Farmer and Martin R. Zirnbauer},
  journal={arXiv: Mathematical Physics},
For the classical compact Lie groups K = U(N) the autocorrelation functions of ratios of random characteristic polynomials are studied. Basic to our treatment is a property shared by the spinor representation of the spin group with the Shale-Weil representation of the metaplectic group: in both cases the character is the analytic square root of a determinant or the reciprocal thereof. By combining this fact with Howe's theory of supersymmetric dual pairs (g,K), we express the K-Haar average… 
A Symplectic Test of the L-Functions Ratios Conjecture
Recently Conrey, Farmer and Zirnbauer [8, 9] conjectured formulas for the averages over a family of ratios of products of shifted L-functions. Their L-functions Ratios Conjecture predicts both the
Auto-correlation functions for unitary groups
. We compute the auto-correlations functions of order m ≥ 1 for the characteristic polynomials of random matrices from certain subgroups of the unitary groups U(2) and U(3) by applying branching
A few remarks on colour–flavour transformations, truncations of random unitary matrices, Berezin reproducing kernels and Selberg-type integrals
We investigate diverse relations of the colour–flavour transformations (CFT) introduced by Zirnbauer in 'Supersymmetry for systems with unitary disorder: circular ensembles' (1996 J. Phys. A: Math.
Some Results in the Theory of Low-Lying Zeros of Families of L-Functions
While Random Matrix Theory has successfully modeled the limiting behavior of many quantities of families of L-functions, especially the distributions of zeros and values, the theory frequently cannot
Mixed moments of characteristic polynomials of random unitary matrices
Following the work of Conrey, Rubinstein and Snaith and Forrester and Witte we examine a mixed moment of the characteristic polynomial and its derivative for matrices from the unitary group U(N)
Correlations of eigenvalues and Riemann zeros
Interest in comparing the statistics of the zeros of the Riemann zeta function with random matrix theory dates back to the 1970s and the work of Montgomery and Dyson. Twelve years ago Rudnick and


Random Matrix Theory and ζ(1/2+it)
Abstract: We study the characteristic polynomials Z(U, θ) of matrices U in the Circular Unitary Ensemble (CUE) of Random Matrix Theory. Exact expressions for any matrix size N are derived for the
Autocorrelation of Random Matrix Polynomials
Abstract: We calculate the autocorrelation functions (or shifted moments) of the characteristic polynomials of matrices drawn uniformly with respect to Haar measure from the groups U(N), O(2N) and
Averages of ratios of characteristic polynomials for the compact classical groups
Averages of ratios of characteristic polynomials for the compact classical groups are evaluated in terms of determinants whose dimensions are independent of the matrix rank. These formulas are shown
An exact formula for general spectral correlation function of random Hermitian matrices
We have found an exact formula expressing a general correlation function containing both products and ratios of characteristic polynomials of random Hermitian matrices. The answer is given in the
Unitary Representations of Super Lie Groups and Applications to the Classification and Multiplet Structure of Super Particles
It is well known that the category of super Lie groups (SLG) is equivalent to the category of super Harish-Chandra pairs (SHCP). Using this equivalence, we define the category of unitary
Heat Kernels and Dirac Operators
The past few years have seen the emergence of new insights into the Atiyah-Singer Index Theorem for Dirac operators. In this book, elementary proofs of this theorem, and some of its more recent
Supersymmetry for systems with unitary disorder: circular ensembles
A generalized Hubbard - Stratonovitch transformation relating an integral over random unitary matrices to an integral over Efetov's unitary -model manifold, is introduced. This transformation adapts
Introduction to Superanalysis
1. Grassmann Algebra.- 2. Superanalysis.- 3. Linear Algebra in Z2-Graded Spaces.- 4. Supermanifolds in General.- 5. Lie Superalgebras.- 1. Lie Superalgebras.- 2. Lie Supergroups.- 3. Laplace-Casimir
Autocorrelation of ratios of L-functions
We give a new heuristic for all of the main terms in the quotient of products of L-functions averaged over a family. These conjectures generalize the recent conjectures for mean values of