Unitary and Complex Matrix Models as 1-d Type 0 Strings

We propose that the double scaling behavior of the unitary matrix models, and that of the complex matrix models, is related to type 0B and 0A fermionic string theories. The particular backgrounds involved correspond to < 1 matter coupled to super-Liouville theory. We examine in detail the = 0 or pure supergravity case, which is related to the double scaling limit around the Gross-Witten transition, and find that reversing the sign of the Liouville superpotential interchanges the 0A and 0B theories. We also find smooth transitions between weakly coupled string backgrounds with D-branes, and backgrounds with Ramond-Ramond fluxes only. Finally, we discuss matrix models with multicritical potentials that are conjectured to correspond to 0A/0B string theories based on (2,4k) super-minimal models.Dedicated to Freeman Dyson, a pioneer of large N matrix models, on the occasion of his eightieth birthdayAcknowledgement We would like to thank E. Martinec and E. Witten for useful discussions, and C. Johnson for correspondence. IRK is grateful to the Institute for Advanced Study for hospitality during his work on this paper. The research of JM and NS is supported in part by DOE grant DE-FG02-90ER40542. The research of IRK is supported in part by NSF grant PHY-0243680. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.     

Sigma Models and Minimal Surfaces

Correspondence is established between sigma models, minimal surfaces and the Monge–Ampére equation. The Lax pairs of the minimality condition of the minimal surfaces and the Monge–Ampére equations are given. Existence of infinitely many nonlocal conservation laws is shown and some Bäcklund transformations are also given.     

Quantum Motion on 2D Surfaces of Spherical Topology

When the motion of a particle is constrained on the two-dimensional surface, excess terms exist in usual kinetic energy 1/(2μ) ∑ p _i ² with hermitian form of Cartesian momentum p _i (i = 1,2,3), and the operator ordering should be taken into account in the kinetic energy which turns out to be 1/(2μ) ∑ (1/f _i )p _i f _i p _i where the functions f _i are dummy factors in classical mechanics and nontrivial in quantum mechanics. In this article, the explicit forms of the dummy functions f _i for quantum motion on some 2D surfaces of revolution of spherical topology are given. PACS numbers: 03.65.-w Quantum mechanics, 04.60.Ds Canonical quantization.     

Local Statistics of Realizable Vertex Models

We study planar “vertex” models, which are probability measures on edge subsets of a planar graph, satisfying certain constraints at each vertex, examples including the dimer model, and 1-2 model, which we will define. We express the local statistics of a large class of vertex models on a finite hexagonal lattice as a linear combination of the local statistics of dimers on the corresponding Fisher graph, with the help of a generalized holographic algorithm. Using an n × n torus to approximate the periodic infinite graph, we give an explicit integral formula for the free energy and local statistics for configurations of the vertex model on an infinite bi-periodic graph. As an example, we simulate the 1-2 model by the technique of Glauber dynamics.     

Dijkgraaf–Vafa Matrix Models and Integrable Systems

We review the Dijkgraaf-Vafa correspondence and give some mathematical details of the associated matrix contour integrals, before discussing recent work relating supersymmetric vacua to integrable systems.     

Power-Counting Theorem for Non-Local Matrix Models and Renormalisation

Solving the exact renormalisation group equation à la Wilson-Polchinski perturbatively, we derive a power-counting theorem for general matrix models with arbitrarily non-local propagators. The power-counting degree is determined by two scaling dimensions of the cut-off propagator and various topological data of ribbon graphs. As a necessary condition for the renormalisability of a model, the two scaling dimensions have to be large enough relative to the dimension of the underlying space. In order to have a renormalisable model one needs additional locality properties—typically arising from orthogonal polynomials—which relate the relevant and marginal interaction coefficients to a finite number of base couplings. The main application of our power-counting theorem is the renormalisation of field theories on noncommutative ^D in matrix formulation.Acknowledgement We are grateful to José Gracia-Bondía and Edwin Langmann for discussions concerning the integral representation of the -product and its matrix base. We would like to thank Thomas Krajewski for advertising the Polchinski equation to us and Volkmar Putz for the accompanying study of Polchinskis original proof. We are grateful to Christoph Kopper for indicating to us a way to reduce in our original power-counting estimation the polynomial in to a polynomial in thus permitting immediately the limit ₀. We would also like to thank Manfred Schweda and his group for enjoyable collaboration. We are indebted to the Erwin Schrödinger Institute in Vienna, the Max-Planck-Institute for Mathematics in the Sciences in Leipzig and the Institute for Theoretical Physics of the University of Vienna for the generous support of our collaboration.     

Bulk Universality and Related Properties of Hermitian Matrix Models

We give a new proof of universality properties in the bulk of spectrum of the hermitian matrix models, assuming that the potential that determines the model is globally C ² and locally C ³ function (see Theorem 3.1). The proof as our previous proof in (Pastur and Shcherbina in J. Stat. Phys. 86:109–147, 1997) is based on the orthogonal polynomial techniques but does not use asymptotics of orthogonal polynomials. Rather, we obtain the sin -kernel as a unique solution of a certain non-linear integro-differential equation that follows from the determinant formulas for the correlation functions of the model. We also give a simplified and strengthened version of paper (Boutet de Monvel, et al. in J. Stat. Phys. 79:585–611, 1995) on the existence and properties of the limiting Normalized Counting Measure of eigenvalues. We use these results in the proof of universality and we believe that they are of independent interest.     

Matrix De Rham Complex and Quantum A-infinity algebras

I establish the relation of the non-commutative BV-formalism with super-invariant matrix integration. In particular, the non-commutative BV-equation, defining the quantum A _∞-algebras, introduced in Barannikov (Modular operads and non-commutative Batalin–Vilkovisky geometry. IMRN, vol. 2007, rnm075. Max Planck Institute for Mathematics 2006–48, 2007), is represented via de Rham differential acting on the supermatrix spaces related with Bernstein–Leites simple associative algebras with odd trace q(N), and gl(N|N). I also show that the matrix Lagrangians from Barannikov (Noncommutative Batalin–Vilkovisky geometry and matrix integrals. Isaac Newton Institute for Mathematical Sciences, Cambridge University, 2006) are represented by equivariantly closed differential forms.     

Three Solvable Matrix Models of a Quantum Catastrophe

Three classes of finite-dimensional models of quantum systems exhibiting spectral degeneracies called quantum catastrophes are described in detail. Computer-assisted symbolic manipulation techniques are shown unexpectedly efficient for the purpose.     

Comparison between Highly Complex Location Models and GAMLSS

This paper presents a discussion regarding regression models, especially those belonging to the location class. Our main motivation is that, with simple distributions having simple interpretations, in some cases, one gets better results than the ones obtained with overly complex distributions. For instance, with the reverse Gumbel (RG) distribution, it is possible to explain response variables by making use of the generalized additive models for location, scale, and shape (GAMLSS) framework, which allows the fitting of several parameters (characteristics) of the probabilistic distributions, like mean, mode, variance, and others. Three real data applications are used to compare several location models against the RG under the GAMLSS framework. The intention is to show that the use of a simple distribution (e.g., RG) based on a more sophisticated regression structure may be preferable than using a more complex location model.     

Saturation of Electrostatic Potential: Exactly Solvable 2D Coulomb Models

We test the concepts of renormalized charge and potential saturation, introduced within the framework of highly asymmetric Coulomb mixtures, on exactly solvable Coulomb models. The object of study is the average electrostatic potential induced by a unique “guest” charge immersed in a classical electrolyte, the whole system being in thermal equilibrium at some inverse temperature β. The guest charge is considered to be either an infinite hard wall carrying a uniform surface charge or a charged colloidal particle. The systems are treated as two-dimensional; the electrolyte is modelled by a symmetric two-component plasma (TCP) of point-like ±e charges with logarithmic Coulomb interactions. Two cases are solved exactly: the Debye–Hückel limit β e²→ 0 and the Thirring free-fermion point β e²=2. The results at the free-fermion point can be summarized as follows: (i) The induced electrostatic potential exhibits the asymptotic behavior, at large distances from the guest charge, whose form is different from that obtained in the Debye–Hückel (linear Poisson–Boltzmann) theory. This means that the concept of renormalized charge, developed within the nonlinear Poisson–Boltzmann (PB) theory to describe the screening effect of the electrolyte cloud, fails at the free-fermion point. (ii) In the limit of an infinite bare charge, the induced electrostatic potential saturates at a finite value in every point of the electrolyte region. This fact confirms the previously proposed hypothesis of potential saturation.     

Orthogonal and Symplectic Matrix Models: Universality and Other Properties

We study orthogonal and symplectic matrix models with polynomial potentials and multi interval supports of the equilibrium measures. For these models we find the bounds (similar to those for the hermitian matrix models) for the rate of convergence of linear eigenvalue statistics and for the variance of linear eigenvalue statistics and find the logarithms of partition functions up to the order O(1). We prove also the universality of local eigenvalue statistics in the bulk.     

Jordan-Wigner Approach to Dynamic Correlations in 2D Spin-1/2 Models

We discuss the dynamic properties of the square-lattice spin-1/2 XY model obtained using the two-dimensional Jordan-Wigner fermionization approach. We argue the relevance of the fermionic picture for interpreting the neutron scattering measurements in the two-dimensional frustrated quantum magnet Cs₂CuCl₄.     

Planelike Interfaces in Long-Range Ising Models and Connections with Nonlocal Minimal Surfaces

This paper contains three types of results:

• the construction of ground state solutions for a long-range Ising model whose interfaces stay at a bounded distance from any given hyperplane,
• the construction of nonlocal minimal surfaces which stay at a bounded distance from any given hyperplane,
• the reciprocal approximation of ground states for long-range Ising models and nonlocal minimal surfaces.

In particular, we establish the existence of ground state solutions for long-range Ising models with planelike interfaces, which possess scale invariant properties with respect to the periodicity size of the environment. The range of interaction of the Hamiltonian is not necessarily assumed to be finite and also polynomial tails are taken into account (i.e. particles can interact even if they are very far apart the one from the other). In addition, we provide a rigorous bridge between the theory of long-range Ising models and that of nonlocal minimal surfaces, via some precise limit result.

     

Spatial Statistics and Information Geometry for Parametric Statistical Models of Galaxy Clustering

Poisson spatial processes of points and ofextended objects representing smoothed clusters ofgalaxies are considered; some results are obtained forplanar representations of random filaments, which may help interpret the findings of the Las CampanasRedshift Survey. Based on a model for the voidprobability function, a family of gamma-relateddistributions is investigated as a three-dimensionalmodel for the clustering of galaxies. The unclusteredmodels in this family correspond to the random case andto maximum information-theoretic entropy. The Riemannianinformation metric and Gaussian curvature are derived for the parameter space of thefamily of models, which provides a background on whichto write dynamics for cluster evolution.     

强反射复杂表面随机缺陷检测照明系统分析

分析了采用CCD光电检测法对强反射复杂表面随机缺陷进行检测时的光学照明问题。给出了缺陷显现力的具体定义。对比了平行光和均匀散射光对消除强反射光的不同作用及对不同表面缺陷的显现力。分析了使用均匀散射光作为光源时，工件表面照度对成像质量的影响。给出了在上述条件下光照系统设计的基本原则。     

Semiclassical Orthogonal Polynomials, Matrix Models and Isomonodromic Tau Functions

The differential systems satisfied by orthogonal polynomials with arbitrary semiclassical measures supported on contours in the complex plane are derived, as well as the compatible systems of deformation equations obtained from varying such measures. These are shown to preserve the generalized monodromy of the associated rank-2 rational covariant derivative operators. The corresponding matrix models, consisting of unitarily diagonalizable matrices with spectra supported on these contours are analyzed, and it is shown that all coefficients of the associated spectral curves are given by logarithmic derivatives of the partition function or, more generally, the gap probabilities. The associated isomonodromic tau functions are shown to coincide, within an explicitly computed factor, with these partition functions. Research supported in part by the Natural Sciences and Engineering Research Council of Canada, the Fonds FCAR du Québec and EC ITH Network HPRN-CT-1999-000161.     

Equilibration of Small and Large Subsystems in Field Theories and Matrix Models

Integrable quantum field theories in 1+1 dimensions have recently become amenable to a rigorous construction, but many questions about the structure of their local observables remain open. Our goal is to characterize these local observables in terms of their expansion coefficients in a series expansion by interacting annihilators and creators, similar to form factors. We establish a rigorous one-to-one characterization, where locality of an observable is reflected in analyticity properties of its expansion coefficients; this includes detailed information about the high-energy behaviour of the observable and the growth properties of the analytic functions. Our results hold for generic observables, not only smeared pointlike fields, and the characterizing conditions depend only on the localization region—we consider wedges and double cones—and on the permissible high energy behaviour.