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In recent studies of many-body localization in nonintegrable quantum systems, the distribution of the ratio of two consecutive energy level spacings, $r_n=(E_{n+1}-E_n)/(E_{n}-E_{n-1})$ or $\tilde{r}_n=\min(r_n,r_n^{-1})$, has been used as a measure to quantify the chaoticity, alternative to the more conventional distribution of the level spacings, $s_n=\bar{\rho}(E_n)(E_{n+1}-E_n)$, as the former makes unnecessary the unfolding required for the latter. Based on our previous work on the Tracy-Widom approach to the Janossy densities, we present analytic expressions for the joint probability distribution of two consecutive eigenvalue spacings and the distribution of their ratio for the Gaussian unitary ensemble (GUE) of random Hermitian $N\times N$ matrices at $N\to \infty$, in terms of a system of differential equations. As a showcase of the efficacy of our results for characterizing an approach to quantum chaoticity, we contrast them to arguably the most ideal of all quantum-chaotic spectra: the zeroes of the Riemann $\zeta$ function on the critical line at increasing heights., Comment: 9 pages, 6 figures; (v2) version published in PTEP, subtitle changed

The J\'{a}nossy density for a determinantal point process is the probability density that an interval $I$ contains exactly $p$ points except for those at $k$ designated loci. The J\'{a}nossy density associated with an integrable kernel $\mathbf{K}\doteq (\varphi(x)\psi(y)-\psi(x)\varphi(y))/(x-y)$ is shown to be expressed as a Fredholm determinant $\mathrm{Det}(\mathbb{I}-\tilde{\mathbf{K}}|_I)$ of a transformed kernel $\tilde{\mathbf{K}}\doteq (\tilde{\varphi}(x)\tilde{\psi}(y)-\tilde{\psi}(x)\tilde{\varphi}(y))/(x-y)$. We observe that $\tilde{\mathbf{K}}$ satisfies Tracy and Widom's criteria if $\mathbf{K}$ does, because of the structure that the map $(\varphi, \psi)\mapsto (\tilde{\varphi}, \tilde{\psi})$ is a meromorphic $\mathrm{SL}(2,\mathbb{R})$ gauge transformation between covariantly constant sections. This observation enables application of the Tracy--Widom method to J\'{a}nossy densities, expressed in terms of a solution to a system of differential equations in the endpoints of the interval. Our approach does not explicitly refer to isomonodromic systems associated with Painlev\'{e} equations employed in the preceding works. As illustrative examples we compute J\'{a}nossy densities with $k=1, p=0$ for Airy and Bessel kernels, related to the joint distributions of the two largest eigenvalues of random Hermitian matrices and of the two smallest singular values of random complex matrices., Comment: 18 pages, 8 figs, 2 Mathematica nb's attached. (v2) Additional remarks on SL(r) extension etc, improved agreement with quadrature approx. Version to appear in PTEP

We compute individual distributions of low-lying eigenvalues of massive chiral random matrix ensembles by the Nystr\"om-type quadrature method for evaluating the Fredholm determinant and Pfaffian that represent the analytic continuation of the Janossy densities (conditional gap probabilities). A compact formula for individual eigenvalue distributions suited for precise numerical evaluation by the Nystr\"om-type method is obtained in an explicit form, and the $k^{\rm{\small th}}$ smallest eigenvalue distributions are numerically evaluated for chiral unitary and symplectic ensembles in the microscopic limit. As an application of our result, the low-lying Dirac spectra of the SU(2) lattice gauge theory with $N_F=8$ staggered flavors are fitted to the numerical prediction from the chiral symplectic ensemble, leading to a precise determination of the chiral condensateof a two-color QCD-like system in the future., Comment: 47 pages, 12 figures, 3 tables; 1 mathematica notebook attached; v2: minor corrections, references added; v3: minor corrections in sections 2.2 and 2.3

We compute individual distributions of low-lying eigenvalues of a chiral random matrix ensemble interpolating symplectic and unitary symmetry classes by the Nystr\"om-type method of evaluating the Fredholm Pfaffian and resolvents of the quaternion kernel. The one-parameter family of these distributions are shown to fit excellently the Dirac spectra of SU(2) lattice gauge theory with a constant U(1) background or dynamically fluctuating U(1) gauge field, which weakly breaks the pseudo-reality of the unperturbed SU(2) Dirac operator. Observed linear dependence of the crossover parameter with the strength of U(1) perturbations leads to precise determination of the pseudo-scalar decay constant, as well as the chiral condensate in the effective chiral Lagrangian of AI class., Comment: 28 pages in PTPTeX, 25 figures

Based on the exact relationship to random matrix theory, we present an alternative method of evaluating the probability distribution of the k-th smallest Dirac eigenvalue in the epsilon-regime of QCD and QCD-like theories. By utilizing the Nystrom-type discretization of Fredholm determinants and Pfaffians, practical trouble of evaluating multiple integrations is circumvented and technical restrictions on the parities of the number of flavors and of the topological charge present in our previous treatment for beta=1 and 4 cases [Phys. Rev. D 63, 045012 (2001)] are partly lifted. This method is also applied to the distributions of spacings between k-th nearest-neighboring levels in the mobility edges of Anderson Hamiltonian and Dirac operator in high-temperature QCD., Comment: 7 pages, 10 figures, presented at the 33rd International Symposium on Lattice Field Theory (Lattice 2015), 14-18 July 2015, Kobe, Japan; (v2) fixed a typo in (2.6). PoS LATTICE 2015, 057

We compute statistical distributions of individual low-lying eigenvalues of random matrix ensembles interpolating chiral Gaussian symplectic and unitary ensembles. To this aim we use the Nystrom-type discretization of Fredholm Pfaffians and resolvents of the dynamical Bessel kernel containing a single crossover parameter \rho. The \rho-dependent distributions of the four smallest eigenvalues are then used to fit the Dirac spectra of modulated SU(2) lattice gauge theory, in which the reality of the staggered SU(2) Dirac operator is weakly violated either by the U(1) gauge field or by a constant background flux. Combined use of individual eigenvalue distributions is effective in reducing statistical errors in \rho; its linear dependence on the imaginary chemical potential \mu_I enables precise determination of the pseudo-scalar decay constant F of the SU(2) gauge theory from a small lattice. The U(1)-coupling dependence of an equivalent of F^2 \mu_I^2 in the SU(2) x U(1) theory is also obtained., Comment: 7 pages, 10 figures, presented at the 32nd International Symposium on Lattice Field Theory - Lattice 2014, June 23-28, 2014, Columbia University, New York, NY; (v2) replaces the 3rd row of Table 1 and Figure 3 left that contained errors

Canonical partition functions and Lee-Yang zeros of QCD at finite density and high temperature are studied. Recent lattice simulations have confirmed that the free energy of QCD is a quartic function of quark chemical potential at temperature slightly above pseudo-critical temperature $T_c$, as in the case with a gas of free massless fermions. We present analytic derivation of the canonical partition functions and Lee-Yang zeros for this type of free energy using the saddle point approximation. We also perform lattice QCD simulation in a canonical approach using the fugacity expansion of the fermion determinant, and carefully examine its reliability. By comparing the analytic and numerical results, we conclude that the canonical partition functions follow the Gaussian distribution of the baryon number, and the accumulation of Lee-Yang zeros of these canonical partition functions exhibit the first-order Roberge-Weiss phase transition. We discuss the validity and applicable range of the result and its implications both for theoretical and experimental studies., Comment: 11 pages, 7 figures. Accepted to Phys. Rev. D

We present an analytic expression of the nonperturbative free energy of a double-well supersymmetric matrix model in its double scaling limit, which corresponds to two-dimensional type IIA superstring theory on a nontrivial Ramond-Ramond background. To this end we draw upon the wisdom of random matrix theory developed by Tracy and Widom, which expresses the largest eigenvalue distribution of unitary ensembles in terms of a Painleve II transcendent. Regularity of the result at any value of the string coupling constant shows that the third-order phase transition between a supersymmetry-preserving phase and a supersymmetry-broken phase, previously found at the planar level, becomes a smooth crossover in the double scaling limit. Accordingly, the supersymmetry is always broken spontaneously as its order parameter stays nonzero for the whole region of the coupling constant. Coincidence of the result with the unitary one-matrix model suggests that one-dimensional type 0 string theories partially correspond to the type IIA superstring theory. Our formulation naturally allows for introduction of an instanton chemical potential, and reveals the presence of a novel phase transition, possibly interpreted as condensation of instantons., Comment: 25 pages, 4 figures. v2: new subsection 4.3 and references added. version to be published in JHEP

We examine statistical fluctuation of eigenvalues from the near-edge bulk of QCD Dirac spectra above the critical temperature. For completeness we start by reviewing on the spectral property of Anderson tight-binding Hamiltonians as described by nonlinear sigma models and random matrices, and on the scale-invariant intermediate spectral statistics at the mobility edge. By fitting the level spacing distributions, deformed random matrix ensembles which model multifractality of the wave functions typical of the Anderson localization transition, are shown to provide an excellent effective description for such a critical statistics. Next we carry over the above strategy for the Anderson Hamiltonians to the Dirac spectra. For the staggered Dirac operators of QCD with 2+1 flavors of dynamical quarks at the physical point and of SU(2) quenched gauge theory, we identify the precise location of the mobility edge as the scale-invariant fixed point of the level spacing distribution. The eigenvalues around the mobility edge are shown to obey critical statistics described by the aforementioned deformed random matrix ensembles of unitary and symplectic classes. The best-fitting deformation parameter for QCD at the physical point turns out to be consistent with the Anderson Hamiltonian in the unitary class. Finally, we propose a method of locating the mobility edge at the origin of QCD Dirac spectrum around the critical temperature, by the use of individual eigenvalue distributions of deformed chiral random matrices., Comment: 15 pages, 15 figures, presented at the 31st International Symposium on Lattice Field Theory (Lattice 2013), 29 July - 3 August 2013, Mainz, Germany

Schierenberg et al. [Phys. Rev. E 85, 061130 (2012)] recently applied the Wigner surmise, i.e., substitution of \infty \times \infty matrices by their 2 \times 2 counterparts for the computation of level spacing distributions, to random matrix ensembles in transition between two universality classes. I examine the accuracy and the range of validity of the surmise for the crossover between the Gaussian orthogonal and unitary ensembles by contrasting them with the large-N results that I evaluated using the Nystrom-type method for the Fredholm determinant. The surmised expression at the best-fitting parameter provides a good approximation for 0 \lesssim s \lesssim 2, i.e., the validity range of the original surmise., Comment: 3 pages in REVTeX, 10 figures. (v2) Title changed, version to appear in Phys. Rev. E

We evaluate level spacing and smallest eigenvalue distributions of chiral random matrix ensembles transiting from symplectic or orthogonal to unitary symmetry classes with a crossover parameter rho. As expected from the effective sigma model description, these results can be fitted perfectly to the fundamental or adjoint staggered Dirac spectrum of SU(2) quenched lattice gauge theory under the imaginary chemical potential (twisting) mu. The linear dependence of the parameter rho on mu determines the pion decay constant F as its proportionality constant., Comment: 10 pages in PTPTeX, 15 figures. (v2) Uses new data with antiperiodicity in temporal direction, statistical errors updated. (v3) Version to appear in Prog. Theor. Phys. 128 (2012). arXiv admin note: text overlap with arXiv:1208.3452

Motivated by the statistical fluctuation of Dirac spectrum of QCD-like theories subjected to (pseudo)reality-violating perturbations and in the epsilon-regime, we compute the smallest eigenvalue distribution and the level spacing distribution of chiral and non-chiral parametric random matrix ensembles of Dyson-Mehta-Pandey type. To this end we employ the Nystrom method to numerically evaluate the Fredholm Pfaffian of the integral kernel for the chG(O,S)E-chGUE and G(O,S)E-GUE crossover. We confirm the validity and universality of our results by comparing them with several lattice models, namely fundamental and adjoint staggered Dirac spectra of SU(2) quenched lattice gauge theory under the twisted boundary condition (imaginary chemical potential) or perturbed by phase noise. Both in the zero-virtuality region and in the spectral bulk, excellent one-parameter fitting is achieved already on a small 4^4 lattice. Anticipated scaling of the fitting parameter with the twisting phase, mean level spacing, and the system size allows for precise determination of the pion decay (diffusion) constant F in the low-energy effective Lagrangian., Comment: 18 pages in REVTeX, 30 figures. (v2) 40000 configurations used for all parameter sets, statistical errors included in all quantities, erroneous chi^2/dof corrected. (v3) Version to appear in Phys. Rev. D86 (2012)

We study the phase diagram for two-color QCD in three-dimensional spacetime, as a function of temperature and baryon chemical potential, using the low-energy effective Lagrangian approach. We show one-loop renormalizability at zero temperature, and then use the one-loop effective Lagrangian at finite temperature and chemical potential to show that at low temperature there is a critical line separating the normal and diquark phase, with this critical line ending at a tricritical point. This phase structure is qualitatively similar to that found recently by Splittorff et al for two-color QCD in four-dimensional spacetime, although the details are quite different, due to the different symmetries and the different loop and infrared properties of three-dimensional spacetime., Comment: 14 pp, 1 fig

We study the low energy phase structure of SU(2) gauge theories in three-dimensional spacetime, at finite baryon density. The pseudoreality of representations of SU(2) permits an analytic study of a real baryon chemical potential, and the restriction to 3D results in a different global symmetry breaking pattern from the corresponding 4D model studied previously by Kogut et al. We find a second-order phase transition separating the normal phase and the baryon superconducting phase. The chemical potential dependence of condensates and baryon density are computed. We find that the phase structure and the excitation spectrum are essentially the same as in 4D, despite the different symmetry groups, indicating a universality that is rooted in the properties of Riemannian symmetric spaces., Comment: 11 pp, REVTeX4, no figures. (v2) references added and a typo fixed. version to appear in Nucl. Phys. B

The BC-type Calogero-Sutherland model (CSM) is an integrable extension of the ordinary A-type CSM that possesses a reflection symmetry point. The BC-CSM is related to the chiral classes of random matrix ensembles (RMEs) in exactly the same way as the A-CSM is related to the Dyson classes. We first develop the fermionic replica sigma-model formalism suitable to treat all chiral RMEs. By exploiting ''generalized color-flavor transformation'' we then extend the method to find the exact asymptotics of the BC-CSM density profile. Consistency of our result with the c=1 Gaussian conformal field theory description is verified. The emerging Friedel oscillations structure and sum rules are discussed in details. We also compute the distribution of the particle nearest to the reflection point., Comment: 12 pages, no figure, REVTeX4. sect.V updated, references added (v3)

We analyze the effect of heavy fundamentally charged particles on the finite temperature deconfining phase transition in the 2+1 dimensional Georgi-Glashow model. We show that in the presence of fundamental matter the transition turns into a crossover. The near critical theory is mapped onto the 2 dimensional Ising model in an external magnetic field. Using this mapping we determine the width of the crossover region as well as the specific heat as a function of the fundamental mass., Comment: 7 pp, REVTeX4. Comment and references added (v2)

We employ the fermionic and bosonic replicated nonlinear sigma models to treat Ginibre unitary, symplectic, and orthogonal ensembles of non-Hermitian random matrix Hamiltonians. Using saddle point approach combined with Borel resummation procedure we derive the exact large-N results for microscopic density of states in all three ensembles. We also obtain tails of the density of states as well the two-point function for the unitary ensemble., Comment: REVTeX 3.1, 13 pages, 1 figure; typos fixed (v2)

We derive nonlinear sigma models (chiral Lagrangians) over symmetric spaces U(n), U(2n)/Sp(2n), and U(2n)/O(2n) from U(N), O(N), and Sp(2N) lattice gauge theories coupled to n flavors of staggered fermions, in the large-N and g^2 N limit. To this end, we employ Zirnbauer's color-flavor transformation. We prove the spatial homogeneity of the vacuum configurations of mesons by explicitly solving the large-N saddle point equations, and thus establish the above patterns of spontaneous chiral symmetry breaking without any assumptions., Comment: REVTeX 3.1, 4 pages, no figure

We consider orthogonal, unitary, and symplectic ensembles of random matrices with (1/a)(ln x)^2 potentials, which obey spectral statistics different from the Wigner-Dyson and are argued to have multifractal eigenstates. If the coefficient $a$ is small, spectral correlations in the bulk are universally governed by a translationally invariant, one-parameter generalization of the sine kernel. We provide analytic expressions for the level spacing distribution functions of this kernel, which are hybrids of the Wigner-Dyson and Poisson distributions. By tuning the single parameter, our results can be excellently fitted to the numerical data for three symmetry classes of the three-dimensional Anderson Hamiltonians at the metal-insulator transition, previously measured by several groups using exact diagonalization., Comment: 12 pages, 8 figures, REVTeX. Additional figure and text on the level number variance, to appear in Phys.Rev.E

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., Comment: 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)

I shall present a proof of universality of the microscopic spectral correlations in Verbaarschot's random matrix models of QCD, to corroborate the beautiful agreement between the predictions from the gaussian model and the numerical data. Rather than following closely the original proof, I shall disguise it with the conventional approach of Q, P operators and with the method employed in an alternative proof by Kanzieper-Freilikher, in a hope that borrowing notions from quantum mechanics may add the proof some pedagogical flavor. I shall also discuss a problem associated with the multi-criticality., Comment: 9 pages, LaTeX2e + plenum.sty. To appear in the Proceedings of the NATO Workshop "New Developments in Quantum Field Theory", Zakopane, June 1997

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)., Comment: 5 pages, REVTeX. Misprint corrected