Level Spacing Distribution of Critical Random Matrix Ensembles

Abstract

We consider unitary invariant random matrix ensembles which obey spectral statistics different from the Wigner-Dyson, including unitary ensembles with slowly (~(log x)^2) growing potentials and the finite-temperature fermi gas model. If the deformation parameters in these matrix ensembles are small, the asymptotically translational-invariant region in the spectral bulk is universally governed by a one-parameter generalization of the sine kernel. We provide an analytic expression for the distribution of the eigenvalue spacings of this universal asymptotic kernel, which is a hybrid of the Wigner-Dyson and the Poisson distributions, by determining the Fredholm determinant of the universal kernel in terms of a Painleve VI transcendental function.

5 pages, 1 figure, REVTeX; restriction on the parameter stressed, figure replaced, refs added (v2); typos (factors of pi) in (35), (36) corrected (v3); minor changes incl. title, version to appear in Phys.Rev.E (v4)

References in corpus (5)

• Random Matrix Theories in Quantum Physics: Common Concepts
• New Class of Random Matrix Ensembles with Multifractal Eigenvectors
• Microscopic universality in the spectrum of the lattice Dirac operator
• Multicritical Microscopic Spectral Correlators of Hermitian and Complex Matrices
• Novel universal correlations in invariant random-matrix models

Cited by in corpus (10)

• Level spacings at the metal-insulator transition in the Anderson Hamiltonians and multifractal random matrix ensembles
• Poisson to Random Matrix Transition in the QCD Dirac Spectrum
• Chiral Random Matrix Model for Critical Statistics
• Localization of Dirac Fermions in Finite-Temperature Gauge Theory
• Critical level statistics and anomalously localized states at the Anderson transition
• Universality crossover between chiral random matrix ensembles and twisted SU(2) lattice Dirac spectra
• On the multifractal dimensions and statistical properties of critical ensembles characterized by the three classical Wigner-Dyson symmetry classes
• Parametric level statistics in random matrix theory: Exact solution
• Geometry of the Ising persistence problem and the universal Bonnet-Manin Painlevé VI distribution
• A Note on Wiener-Hopf Determinants and the Borodin-Okounkov Identity