- •Preface
- •Foreword
- •The Henri Poincaré Prize
- •Contributors
- •Contents
- •Stability of Doubly Warped Product Spacetimes
- •Introduction
- •Warped Product Spacetimes
- •Asymptotic Behavior
- •Fuchsian Method
- •Velocity Dominated Equations
- •Velocity Dominated Solution
- •Stability
- •References
- •Introduction
- •The Tomonaga Model with Infrared Cutoff
- •The RG Analysis
- •The Dyson Equation
- •The First Ward Identity
- •The Second Ward Identity
- •The Euclidean Thirring Model
- •References
- •Introduction
- •Lie and Hopf Algebras of Feynman Graphs
- •From Hochschild Cohomology to Physics
- •Dyson-Schwinger Equations
- •References
- •Introduction
- •Quantum Representation and Dynamical Equations
- •Quantum Singularity Problem
- •Examples for Properties of Solutions
- •Effective Theory
- •Summary
- •Introduction
- •Results and Strategy of Proofs
- •References
- •Introduction
- •Critical Scaling Limits and SLE
- •Percolation
- •The Critical Loop Process
- •General Features
- •Construction of a Single Loop
- •The Near-Critical Scaling Limit
- •References
- •Black Hole Entropy Function and Duality
- •Introduction
- •Entropy Function and Electric/Magnetic Duality Covariance
- •Duality Invariant OSV Integral
- •References
- •Weak Turbulence for Periodic NLS
- •Introduction
- •Arnold Diffusion for the Toy Model ODE
- •References
- •Angular Momentum-Mass Inequality for Axisymmetric Black Holes
- •Introduction
- •Variational Principle for the Mass
- •References
- •Introduction
- •The Trace Map
- •Introduction
- •Notations
- •Entanglement-Assisted Quantum Error-Correcting Codes
- •The Channel Model: Discretization of Errors
- •The Entanglement-Assisted Canonical Code
- •The General Case
- •Distance
- •Generalized F4 Construction
- •Bounds on Performance
- •Conclusions
- •References
- •Particle Decay in Ising Field Theory with Magnetic Field
- •Ising Field Theory
- •Evolution of the Mass Spectrum
- •Particle Decay off the Critical Isotherm
- •Unstable Particles in Finite Volume
- •References
- •Lattice Supersymmetry from the Ground Up
- •References
- •Stable Maps are Dense in Dimensional One
- •Introduction
- •Density of Hyperbolicity
- •Quasi-Conformal Rigidity
- •How to Prove Rigidity?
- •The Strategy of the Proof of QC-Rigidity
- •Enhanced Nest Construction
- •Small Distortion of Thin Annuli
- •Approximating Non-renormalizable Complex Polynomials
- •References
- •Large Gap Asymptotics for Random Matrices
- •References
- •Introduction
- •Coupled Oscillators
- •Closure Equations
- •Introduction
- •Conservative Stochastic Dynamics
- •Diffusive Evolution: Green-Kubo Formula
- •Kinetic Limits: Phonon Boltzmann Equation
- •References
- •Introduction
- •Bethe Ansatz for Classical Lie Algebras
- •The Pseudo-Differential Equations
- •Conclusions
- •References
- •Kinetically Constrained Models
- •References
- •Introduction
- •Local Limits for Exit Measures
- •References
- •Young Researchers Symposium Plenary Lectures
- •Dynamics of Quasiperiodic Cocycles and the Spectrum of the Almost Mathieu Operator
- •Magic in Superstring Amplitudes
- •XV International Congress on Mathematical Physics Plenary Lectures
- •The Riemann-Hilbert Problem: Applications
- •Trying to Characterize Robust and Generic Dynamics
- •Cauchy Problem in General Relativity
- •Survey of Recent Mathematical Progress in the Understanding of Critical 2d Systems
- •Random Methods in Quantum Information Theory
- •Gauge Fields, Strings and Integrable Systems
- •XV International Congress on Mathematical Physics Specialized Sessions
- •Condensed Matter Physics
- •Rigorous Construction of Luttinger Liquids Through Ward Identities
- •Edge and Bulk Currents in the Integer Quantum Hall Effect
- •Dynamical Systems
- •Statistical Stability for Hénon Maps of Benedics-Carleson Type
- •Entropy and the Localization of Eigenfunctions
- •Equilibrium Statistical Mechanics
- •Short-Range Spin Glasses in a Magnetic Field
- •Non-equilibrium Statistical Mechanics
- •Current Fluctuations in Boundary Driven Interacting Particle Systems
- •Fourier Law and Random Walks in Evolving Environments
- •Exactly Solvable Systems
- •Correlation Functions and Hidden Fermionic Structure of the XYZ Spin Chain
- •Particle Decay in Ising Field Theory with Magnetic Field
- •General Relativity
- •Einstein Spaces as Attractors for the Einstein Flow
- •Loop Quantum Cosmology
- •Operator Algebras
- •From Vertex Algebras to Local Nets of von Neuman Algebras
- •Non-Commutative Manifolds and Quantum Groups
- •Partial Differential Equations
- •Weak Turbulence for Periodic NSL
- •Ginzburg-Landau Dynamics
- •Probability Theory
- •From Planar Gaussian Zeros to Gravitational Allocation
- •Quantum Mechanics
- •Recent Progress in the Spectral Theory of Quasi-Periodic Operators
- •Recent Results on Localization for Random Schrödinger Operators
- •Quantum Field Theory
- •Algebraic Aspects of Perturbative and Non-Perturbative QFT
- •Quantum Field Theory in Curved Space-Time
- •Lattice Supersymmetry From the Ground Up
- •Analytical Solution for the Effective Charging Energy of the Single Electron Box
- •Quantum Information
- •One-and-a-Half Quantum de Finetti Theorems
- •Catalytic Quantum Error Correction
- •Random Matrices
- •Probabilities of a Large Gap in the Scaled Spectrum of Random Matrices
- •Random Matrices, Asymptotic Analysis, and d-bar Problems
- •Stochastic PDE
- •Degenerately Forced Fluid Equations: Ergodicity and Solvable Models
- •Microscopic Stochastic Models for the Study of Thermal Conductivity
- •String Theory
- •Gauge Theory and Link Homologies
Large Gap Asymptotics for Random Matrices |
419 |
and Hankel determinants. We then analyse the related systems of orthogonal polynomials, and use identities for the logarithmic derivatives of Toeplitz and Hankel determinants. As a byproduct of this approach, we obtain the asymptotics for the orthogonal polynomials (for weights fα and wα ).
References
1.J. Baik, R. Buckingham, and J. DiFranco, Asymptotics of Tracy-Widom distributions and the total integral of a Painlevé II function. Commun. Math. Phys. 280, 463–497 (2008)
2.P. Deift and X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problem. Ann. Math. 137, 295–368 (1993)
3.P. Deift, A. Its, and X. Zhou, A Riemann-Hilbert approach to asymptotic problems arising in the theory of random matrix models, and also in the theory of integrable statistical mechanics. Ann. Math. 146, 149–235 (1997)
4.P. Deift, A. Its, I. Krasovsky, and X. Zhou, The Widom-Dyson constant for the gap probability in random matrix theory. J. Comput. Appl. Math. 202, 26–47 (2007)
5.P. Deift, A. Its, and I. Krasovsky, Asymptotics of the Airy-kernel determinant. Commun. Math. Phys. 278, 643–678 (2008)
6.J. des Cloizeaux and M.L. Mehta, Asymptotic behavior of spacing distributions for the eigenvalues of random matrices. J. Math. Phys. 14, 1648–1650 (1973)
7.F. Dyson, Fredholm determinants and inverse scattering problems. Commun. Math. Phys. 47, 171–183 (1976)
8.T. Ehrhardt, Dyson’s constant in the asymptotics of the Fredholm determinant of the sine kernel. Commun. Math. Phys. 272, 683–698 (2007)
9.A.S. Fokas, A.R. Its, and A.V. Kitaev, The isomonodromy approach to matrix models in 2D quantum gravity. Commun. Math. Phys. 147, 395–430 (1992)
10.I.V. Krasovsky, Gap probability in the spectrum of random matrices and asymptotics of polynomials orthogonal on an arc of the unit circle. Int. Math. Res. Not. 2004, 1249–1272 (2004)
11.M.L. Mehta, Random Matrices. Academic Press, San Diego (1990)
12.C. Tracy and H. Widom, Level-spacing distributions and the Airy kernel. Commun. Math. Phys. 159, 151–174 (1994)
13.H. Widom, The strong Szego˝ limit theorem for circular arcs. Indiana Univ. Math. J. 21, 277– 283 (1971)
14.H. Widom, The asymptotics of a continuous analogue of orthogonal polynomials. J. Approx. Theory 77, 51–64 (1994)