# Random Matrix Theory for the Hermitian Wilson Dirac Operator and the chGUE-GUE Transition

Akemann G, Nagao T (2011)

JHEP 2011(10): 60.

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Akemann, Gernot

^{UniBi}; Nagao, TaroAbstract

We introduce a random two-matrix model interpolating between a chiralHermitian (2n+nu)x(2n+nu) matrix and a second Hermitian matrix withoutsymmetries. These are taken from the chiral Gaussian Unitary Ensemble (chGUE)and Gaussian Unitary Ensemble (GUE), respectively. In the microscopic large-nlimit in the vicinity of the chGUE (which we denote by weakly non-chiral limit)this theory is in one to one correspondence to the partition function of Wilsonchiral perturbation theory in the epsilon regime, such as the related twomatrix-model previously introduced in refs. [20,21]. For a generic number offlavours and rectangular block matrices in the chGUE part we derive aneigenvalue representation for the partition function displaying a Pfaffianstructure. In the quenched case with nu=0,1 we derive all spectral correlationsfunctions in our model for finite-n, given in terms of skew-orthogonalpolynomials. The latter are expressed as Gaussian integrals over standardLaguerre polynomials. In the weakly non-chiral microscopic limit this yieldsall corresponding quenched eigenvalue correlation functions of the HermitianWilson operator.

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Akemann G, Nagao T. Random Matrix Theory for the Hermitian Wilson Dirac Operator and the chGUE-GUE Transition.

*JHEP*. 2011;2011(10):60.Akemann, G., & Nagao, T. (2011). Random Matrix Theory for the Hermitian Wilson Dirac Operator and the chGUE-GUE Transition.

*JHEP*,*2011*(10), 60. doi:10.1007/JHEP10(2011)060Akemann, G., and Nagao, T. (2011). Random Matrix Theory for the Hermitian Wilson Dirac Operator and the chGUE-GUE Transition.

*JHEP*2011, 60.Akemann, G., & Nagao, T., 2011. Random Matrix Theory for the Hermitian Wilson Dirac Operator and the chGUE-GUE Transition.

*JHEP*, 2011(10), p 60. G. Akemann and T. Nagao, “Random Matrix Theory for the Hermitian Wilson Dirac Operator and the chGUE-GUE Transition”,

*JHEP*, vol. 2011, 2011, pp. 60. Akemann, G., Nagao, T.: Random Matrix Theory for the Hermitian Wilson Dirac Operator and the chGUE-GUE Transition. JHEP. 2011, 60 (2011).

Akemann, Gernot, and Nagao, Taro. “Random Matrix Theory for the Hermitian Wilson Dirac Operator and the chGUE-GUE Transition”.

*JHEP*2011.10 (2011): 60.
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