Article
Topology, Random Matrix Theory and the spectrum of the Wilson Dirac operator
12/2011;
Source: arXiv

Article: The Effect of the Low Energy Constants on the Spectral Properties of the Wilson Dirac Operator
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ABSTRACT: We summarize recent analytical results obtained for lattice artifacts of the nonHermitian Wilson Dirac operator. Hereby we discuss the effect of all three low energy constants. In particular we study the limit of small lattice spacing and also consider the regime of large lattice spacing which is closely related to the mean field limit. Thereby we extract simple relations between measurable quantities like the average number of additional real modes and the low energy constants. These relations may improve the fitting for the low energy constants.10/2013;  [Show abstract] [Hide abstract]
ABSTRACT: Random matrix theory has been successfully applied to lattice quantum chromodynamics. In particular, a great deal of progress has been made on the understanding, numerically as well as analytically, of the spectral properties of the WilsonDirac operator. In this paper, we study the infrared spectrum of the WilsonDirac operator via random matrix theory including the three leading order a^{2} correction terms that appear in the corresponding chiral Lagrangian. A derivation of the joint probability density of the eigenvalues is presented. This result is used to calculate the density of the complex eigenvalues, the density of the real eigenvalues, and the distribution of the chiralities over the real eigenvalues. A detailed discussion of these quantities shows how each lowenergy constant affects the spectrum. Especially we consider the limit of small and large (which is almost the mean field limit) lattice spacing. Comparisons with Monte Carlo simulations of the random matrix theory show a perfect agreement with the analytical predictions. Furthermore we present some quantities which can be easily used for comparison of lattice data and the analytical results.Physical Review D 11/2013; 88(9). · 4.86 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: The unitary Wilson random matrix theory is an interpolation between the chiral Gaussian unitary ensemble and the Gaussian unitary ensemble. This new way of interpolation is also reflected in the orthogonal polynomials corresponding to such a random matrix ensemble. Although the chiral Gaussian unitary ensemble as well as the Gaussian unitary ensemble are associated to the Dyson index $\beta=2$ the intermediate ensembles exhibit a mixing of orthogonal polynomials and skeworthogonal polynomials. We consider the Hermitian as well as the nonHermitian Wilson random matrix and derive the corresponding polynomials, their recursion relations, ChristoffelDarbouxlike formulas, Rodrigues formulas and representations as random matrix averages in a unifying way. With help of these results we derive the unquenched $k$point correlation function of the Hermitian and then nonHermitian Wilson random matrix in terms of two flavour partition functions only. This representation is due to a Pfaffian factorization drastically simplifying the expressions for numerical applications. It also serves as a good starting point for studying the WilsonDirac operator in the $\epsilon$regime of lattice quantum chromodynamics.Journal of Physics A Mathematical and Theoretical 02/2012; 45(20). · 1.77 Impact Factor
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