Shinsuke M. Nishigaki

Universality of random matrices in the microscopic limit and the Dirac operator spectrum

Abstract 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.

Universal spectral correlators and massive Dirac operators

We derive the large-N spectral correlators of complex matrix ensembles with weights that in the context of Dirac spectra correspond to N massive fermions, and prove that the results are 0939 universal in the appropriate scaling limits. The resulting microscopic spectral densities satisfy exact spectral sum rules of massive Dirac operators in QCD.

Distribution of the k-th smallest Dirac operator eigenvalue

Based on the exact relationship to Random Matrix Theory, we derive the probability distribution of the k-th smallest Dirac operator eigenvalue in the microscopic finite-volume scaling regime of QCD and related gauge theories.

Smallest Dirac eigenvalue distribution from random matrix theory

Institut fu¨r Theoretische Physik, Technische Universit¨at Mu¨nchen, D-85747 Garching, Germany(March 2, 1998)We derive the hole probability and the distribution of the smallest eigenvalue of chiral hermitianrandom matrices corresponding to Dirac operators coupled to massive quarks in QCD. They areexpressed in terms of the QCD partition function in the mesoscopic regime. Their universality isexplicitly related to that of the microscopic massive Bessel kernel.PACS number(s): 05.45.+b, 12.38.Aw, 12.38.Lg

Multicritical Microscopic Spectral Correlators of Hermitian and Complex Matrices

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.

A nonperturbative theory of randomly branching chains

Abstract We present a nonperturbative theory of randomly branching chains (or polymers). The method is based upon the existence of critical points in the large- N limit of O( N ) symmetric vector models. We derive a class of ordinary differential equations governing the behaviour of the partition function with respect to the coupling constants. Solving these equations explicitly, we derive several exact results. Our model is useful to make comparison with the recent nonperturbative studies of two-dimensional quantum gravity since the structure of the differential equations has many parallel features, such as the flow property, the Virasoro structure in the Schwinger-Dyson equation, and so on, with those of random surfaces.

Universal massive spectral correlators and three-dimensional QCD

Based on random matrix theory in the unitary ensemble, we derive the double-microscopic massive spectral correlators corresponding to the Dirac operator of QCD_3 with an even number of fermions N_f. We prove that these spectral correlators are universal, and demonstrate that they satisfy exact massive spectral sum rules of QCD_3 in a phase where flavor symmetries are spontaneously broken according to U(N_f) -> U(N_f/2) x U(N_f/2).

Renormalization group flow in one and two matrix models

Abstract Large- N renormalization group equations for one- and two-matrix models are derived. The exact renormalization group equation involving infinitely many induced interactions can be rewritten in a form that has a finite number of coupling constants by taking account of reparametrization identities. Despite the nonlinearity of the equation, the location of fixed points and the scaling exponents can be extracted from the equation. They agree with the spectrum of relevant operators in the exact solution. A linearized β-function approximates well the global phase structure which includes several nontrivial fixed points. The global renormalization group flow suggests a kind of c -theorem in two-dimensional quantum gravity.

Nonlinear Renormalization Group Equation for Matrix Models

Abstract An exact renormalization group equation is derived for the free energy of matrix models. The renormalization group equation turns out to be nonlinear for matrix models, as opposed to vector models, where it is linear. An algorithm for determining the critical coupling constant and the critical exponent is obtained. As concrete examples, one-matrix models with one and two coupling constants are analyzed and the exact values of the critical coupling constant and the associated critical exponent are found.

Level spacings at the metal-insulator transition in the Anderson Hamiltonians and multifractal random matrix ensembles

We consider orthogonal, unitary, and symplectic ensembles of random matrices with