We prove the universality of correlation functions of chiral unitary and unitary ensembles of random matrices in the microscopic limit. The essence of the proof consists in reducing the three-term recursion relation for the relevant orthogonal polynomials into a Bessel equation governing the local asymptotics around the origin. The possible physical interpretation as the universality of the soft spectrum of the Dirac operator is briefly discussed

We report on the behavior of the eigenvalue distribution of the Dirac operator in (2+1)-flavor QCD at finite temperature, using the HISQ action. We calculate the eigenvalue density at several values of the temperature close to the pseudo-critical temperature. For this study we use gauge field configurations generated on lattices of size 32 3 × 8 with two light quark masses corresponding to pion masses of about 160 and 115 MeV. We find that the eigenvalue density below T c receives large contributions from near-zero modes which become smaller as the temperature increases or the light quark mass decreases. Moreover we find no clear evidence for a gap in the eigenvalue density up to 1.1T c. We also analyze the eigenvalue density near T c where it appears to show a power-law behavior consistent with what is expected in the critical region near the second order chiral symmetry restoring phase transition in the massless limit.

We derive the large-N spectral correlators of complex matrix ensembles with weights that in the context of Dirac spectra correspond to N f massive fermions, and prove that the results are universal in the appropriate scaling limits. The resulting microscopic spectral densities satisfy exact spectral sum rules of massive Dirac operators in QCD.

We derive the hole probability and the distribution of the smallest eigenvalue of chiral hermitian random matrices corresponding to Dirac operators coupled to massive quarks in QCD. They are expressed in terms of the QCD partition function in the mesoscopic regime. Their universality is explicitly related to that of the microscopic massive Bessel kernel. PACS number(s): 05.45.+b, 12.38.Aw, 12.38.Lg There has long been an attractive idea that the lowenergy physics of a complex system can be described by a simple effective theory which respects the global symmetries of the original system. As an example, the quantum spectral statistics of a classically chaotic system is believed to be described by a random matrix theory belonging to the same universality class as the former [1]. One new manifestation of essentially the same idea is the recent observation that QCD Dirac operator spectra on the scale λ = O(1/V 4 ) (where V 4 is the space-time volume) measured in lattice Monte Carlo simulations [2] are in excellent agreement with the predictions from those large-N random matrix theories [3,4] that share the same global symmetries as QCD. The suitably rescaled (microscopic) spectral correlation functions thus seem to provide exact finite-size scaling functions for QCD in a finite volume. Very recently, the microscopic spectral correlators have been calculated from random matrix theories that include the effect of fermion determinants with masses m ≃ O(1/V 4 ) [5][6][7] (see also [8]). When λ and m are measured in units of the mean level spacing at zero virtuality, all the random matrix predictions turn out to be universal, i.e., insensitive to the details of the random matrix potential [4][5][6]. Although the question of whether or not QCD is included in the same universality class cannot be answered by demonstrating the existence of the wide range of universality within random matrix theories, it provides strong support for the former.From the field-theoretic point of view [9] it would be most surprising if these observables would not also be computable solely within the framework of finite-volume generating functionals (partition functions) for the order parameter ψ ψ . If not, large-N random matrix theory, which in principle is foreign to the pertinent field theory language, would seem to be a new ingredient required to describe the observed spectral correlators. It has recently been shown that a description entirely in terms of finite-volume partition functions is indeed also possible [10].In order to confirm by numerical simulations that the low-lying spectra of QCD Dirac operators can be described alternatively by large-N random matrix theories, it is in practice most convenient to measure the distribution of the smallest eigenvalue [2] and compare that to the random matrix prediction [11,7]. Since the smallest eigenvalue distribution largely consists of the first peak of the microscopic spectral density (see Fig. 1, ζ ≡ N λ), we expect it to be universal. In fact, the proven universality of the massive kerne...

We find the microscopic spectral densities and the spectral correlators associated with multicritical behavior for both hermitian and complex matrix ensembles, and show their universality. We conjecture that microscopic spectral densities of Dirac operators in certain theories without spontaneous chiral symmetry breaking may belong to these new universality classes.