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Published online by Cambridge University Press: 24 October 2025
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- The Toda Lattice and Universality for the Computation of the Eigenvalues of a Random Matrix , pp. 155 - 159Publisher: Cambridge University PressPrint publication year: 2025
References
Abraham, R, and Marsden, JE. 1978. Foundations of Mechanics, Revised, Enlarged, Reset. Benjamin/Cummings. 16, 24Google Scholar
Adler, M. 1979. Trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg-de Vries type equations. Inventiones Mathematicae, 50, 219–248. 32CrossRefGoogle Scholar
Agrotis, M, Damianou, P, and Sophocleous, C. 2006. The Toda lattice is super-integrable. Physica A: Statistical Mechanics and Its Applications, 365, 235–243. 33, 112CrossRefGoogle Scholar
Arnold, VI. 1978. Mathematical Methods of Classical Mechanics. Springer. 16 Bakhtin, Y, and Correll, J. 2012. A neural computation model for decision-making times. Journal of Mathematical Psychology, 56(5), 333–340. 4Google Scholar
Beals, R, and Sattinger, DH. 1991. On the complete integrability of completely integrable systems. Communications in Mathematical Physics, 138(3), 409–436. 30CrossRefGoogle Scholar
Bloemendal, A, Erdős, L, Knowles, A, Yau, H-T, and Yin, J. 2014. Isotropic local laws for sample covariance and generalized Wigner matrices. Electronic Journal of Probability, 19, 1–53. 150Google Scholar
Bourgade, P, and Yau, H-T. 2017. The Eigenvector Moment Flow and local Quantum Unique Ergodicity. Communications in Mathematical Physics, 350(1), 231–278. 122, 125, 126CrossRefGoogle Scholar
Bourgade, P, Erdős, L, and Yau, H-T. 2014. Edge universality of beta ensembles. Communications in Mathematical Physics, 332(1), 261–353. 123, 125, 132CrossRefGoogle Scholar
Chu, MT. 1984. The generalized Toda flow, the QR algorithm, and the Center Manifold Theory. SIAM Journal on Algebraic Discrete Methods 5(2), 187–201. 7CrossRefGoogle Scholar
Chu, MT. 1986. A differential equation approach to the singular value decomposition of bidiagonal matrices. Linear Algebra and Its Applications, 80, 71–80. 14CrossRefGoogle Scholar
Date, E, and Tanaka, S. 1976. Analogue of inverse scattering theory for the discrete Hill’s equation and exact solutions for the periodic Toda lattice. Progress of Theoretical Physics, 55(2), 457–465. 13CrossRefGoogle Scholar
Deift, P. 2000. Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. American Mathematical Society. 122Google Scholar
Deift, P, and Gioev, D. 2007. Universality at the edge of the spectrum for unitary, orthogonal, and symplectic ensembles of random matrices. Communications on Pure and Applied Mathematics, 60(6), 867–910. 125CrossRefGoogle Scholar
Deift, P, and Trogdon, T. 2017. Universality for Eigenvalue algorithms on sample covariance matrices. SIAM Journal on Numerical Analysis, 55(6), 2835–2862. 5, 11, 133CrossRefGoogle Scholar
Deift, P, and Trogdon, T. 2018. Universality for the Toda algorithm to compute the largest Eigenvalue of a random matrix. Communications on Pure and Applied Mathematics, 71(3), 505–536. 5, 11CrossRefGoogle Scholar
Deift, P, Kamvissis, S, Kriecherbauer, T, and Zhou, X. 1975. The Toda rarefaction problem. Communications on Pure and Applied Mathematics, 49(1), 35–83. 133.0.CO;2-8>CrossRefGoogle Scholar
Deift, P, Lund, F, and Trubowitz, E. 1980. Nonlinear wave equations and constrained harmonic motion. Communications in Mathematical Physics, 74(2), 141–188. 24, 26, 102CrossRefGoogle Scholar
Deift, P, Nanda, T, and Tomei, C. 1983. Ordinary differential equations and the symmetric Eigenvalue problem. SIAM Journal on Numerical Analysis, 20, 1–22. 3, 7, 10CrossRefGoogle Scholar
Deift, P, Li, L-C, Nanda, T, and Tomei, C. 1986. The Toda flow on a generic orbit is integrable. Communications on Pure and Applied Mathematics, 39(2), 183–232. 33, 66, 71, 75, 88, 96, 100CrossRefGoogle Scholar
Deift, P, Kriecherbauer, T, McLaughlin, KT-R, Venakides, S, and Zhou, X. 1999. Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Communications on Pure and Applied Mathematics, 52(11), 1335–1425. 1243.0.CO;2-1>CrossRefGoogle Scholar
Deift, PA, Li, L-C, Spohn, H, Tomei, C, and Trogdon, T. 2022. On the open Toda chain with external forcing. Pure and Applied Functional Analysis, 7(3), 915–945. 8, 14Google Scholar
Deift, PA, Menon, G, Olver, S, and Trogdon, T. 2014. Universality in numerical computations with random data. Proceedings of the National Academy of Sciences of the United States of America, 111(42), 14973–14978. 3, 4, 132CrossRefGoogle ScholarPubMed
Ding, X, and Trogdon, T. 2021. A Riemann–Hilbert approach to the perturbation theory for orthogonal polynomials: Applications to numerical linear algebra and random matrix theory. arXiv preprint 2112.12354, 1–77. 151Google Scholar
Dunford, N, and Schwartz, JT. 1988. Linear Operators, Part I: General Theory. Wiley. 69Google Scholar
Egorova, I, Michor, J, Pryimak, A, and Teschl, G. 2023. Long-time asymptotics for Toda shock waves in the modulation region. Journal of Mathematical Physics, Analysis, Geometry, 19(1), 396–442. 13Google Scholar
Erdős, L. 2012. Universality for random matrices and log-gases. Current Developments in Mathematics, 2012(1), 59–132. 126CrossRefGoogle Scholar
Erdős, L, Yau, H-T, and Yin, J. 2012. Rigidity of Eigenvalues of generalized Wigner matrices. Advances in Mathematics, 229(3), 1435–1515. 122, 126CrossRefGoogle Scholar
Erdős, L, Knowles, A, Yau, H-T, and Yin, J. 2013. The local semicircle law for a general class of random matrices. Electronic Journal of Probability, 18, 1–58. 122CrossRefGoogle Scholar
Flaschka, H. 1974a. On the Toda lattice. II: Inverse-scattering solution. Progress of Theoretical Physics, 51(3), 703–716. 8, 13, 32, 37CrossRefGoogle Scholar
Flaschka, H. 1974b. The Toda lattice. I. Existence of integrals. Physical Review B, 9(4), 1924–1925. 8, 13, 32, 37CrossRefGoogle Scholar
Francis, J. 1961. The QR transformation: A unitary analogue to the LR transformation - Part 1. The Computer Journal, 3(4), 265–271. 9CrossRefGoogle Scholar
Fried, D. 1986. The cohomology of an isospectral flow. Proceedings of the American Mathematical Society, 98, 363–368. 121CrossRefGoogle Scholar
Gaifullin, A. 2009. The manifold of isospectral symmetric tridiagonal matrices and realization of cycles by aspherical manifolds. Proceedings of the Steklov Institute of Mathematics, 263, 38–56. 121CrossRefGoogle Scholar
Gardner, CS, Greene, JM, Kruskal, MD, and Miura, RM. 1967. Method for solving the Korteweg–de Vries equation. Physical Review Letters, 19, 1095–1097. 31, 32CrossRefGoogle Scholar
Golub, GH, and Van Loan, CF. 2013. Matrix Computations. Johns Hopkins University Press. 3CrossRefGoogle Scholar
Gragg, WB, and Harrod, WJ. 1984. The numerically stable reconstruction of Jacobi matrices from spectral data. Numerische Mathematik, 44(3), 317–335. 43CrossRefGoogle Scholar
Jiang, T. 2006. How many entries of a typical orthogonal matrix can be approximated by independent normals? The Annals of Probability, 34(4), 1497–1529. 125CrossRefGoogle Scholar
Kac, M, and Van Moerbeke, P. 1975. A complete solution of the periodic Toda problem. Proceedings of the National Academy of Sciences of the United States of America, 72(8), 457–465. 13Google ScholarPubMed
Knowles, A, and Yin, J. 2017. Anisotropic local laws for random matrices. Probability Theory and Related Fields, 169(1–2), 257–352. 150CrossRefGoogle Scholar
Kodama, Y, and Shipman, B. 2018. Fifty years of the finite nonperiodic Toda lattice: a geometric and topological viewpoint. Journal of Physics A: Mathematical and Theoretical, 51353001. 8CrossRefGoogle Scholar
Kostant, B. 1979. The Solution to a generalized Toda Lattice and representation theory. Advances in Mathematics, 34(4), 195–338. 7CrossRefGoogle Scholar
Lax, PD. 1975. Periodic solutions of the KdV equation. Communications on Pure and Applied Mathematics, 28, 141–188. 32CrossRefGoogle Scholar
Leite, R, Saldanha, S, and Tomei, C. 2008. An atlas for tridiagonal isospectral manifolds. Linear Algebra and Applications, 429, 387–402. 51, 104, 113, 117, 118CrossRefGoogle Scholar
Leite, R, Saldanha, S, and Tomei, C. 2010. The asymptotics of Wilkinson’s shift: Loss of cubic convergence. Foundations of Computational Mathematics, 10, 15–36. 34, 118CrossRefGoogle Scholar
Leite, R, Saldanha, S, and Tomei, C. 2013. Dynamics of the symmetric Eigen-value problem with shift strategies. International Mathematics Research Notices, 2013, 4382–4412. 34, 118CrossRefGoogle Scholar
Leite, R, Saldanha, S, Tomei, C, and Torres, DM. 2023. Linearizing Toda and SVD flows on large phase spaces of matrices with real spectrum. Physics B, 450. Paper no. 133752, 10 pp. 14, 104, 113, 118Google Scholar
Li, L-C. 1986. SVD flows on generic symplectic leaves are completely integrable. Advances in Mathematics, 128, 82–118. 14CrossRefGoogle Scholar
Manakov, SV. 1974. Complete integrability and stochasticization of discrete dynamical systems. Soviet Physics JETP, 40(2), 269–274. 8, 13, 32, 37Google Scholar
Manakov, SV. 1976. Remarks on the integration of the Euler equations of an n-dimensional rigid body. Functional Analysis and Applications, 10(4), 328–329. 33CrossRefGoogle Scholar
Monthus, C, and Garel, T. 2013. Typical versus averaged overlap distribution in spin glasses: Evidence for droplet scaling theory. Physical Review B, 88(13), 134204. 126CrossRefGoogle Scholar
Moser, J. 1975a. Finitely many mass points on the line under the influence of an exponential potential – an integrable system. Dynamical Systems, Theory and Applications. Lecture Notes in Physics, vol. 38. Springer, pp. 467–497. 9CrossRefGoogle Scholar
Moser, J. 1975b. Three integrable Hamiltonian systems connected with isospectral deformations. Advances in Mathematics, 16(2), 197–220. 7, 46CrossRefGoogle Scholar
Moser, J. 1981. Integrable Hamiltonian Systems and Spectral Theory. Lezioni Fermi-ane. Accademia Nazionale dei Lincei. 24, 26Google Scholar
Perret, A, and Schehr, G. 2014. Near-extreme Eigenvalues and the first gap of Hermitian random matrices. Journal of Statistical Physics, 156(5), 843–876. 126, 132CrossRefGoogle Scholar
Pfrang, CW, Deift, and Menon, G. 2014. How long does it take to compute the Eigen-values of a random symmetric matrix? Random Matrix Theory, Interacting Particle Systems, and Integrable Systems, MSRI Publications, 65, 411–442. 1, 3CrossRefGoogle Scholar
Ramírez, JA, Rider, B, and Virág, B. 2011. Beta ensembles, stochastic Airy spectrum, and a diffusion. Journal of the American Mathematical Society, 24(4), 919–944. 125, 127CrossRefGoogle Scholar
Reyman, AG, and Semenov-Tian-Shansky, MA. 1979. Reduction of Hamiltonian systems, affine Lie algebras and Lax equations. Inventiones Mathematicae, 54(1), 81–100. 75CrossRefGoogle Scholar
Shcherbina, M. 2009. Edge universality for orthogonal ensembles of random matrices. Journal of Statistical Physics, 136(1), 35–50. 125CrossRefGoogle Scholar
Shub, M, and Vasquez, A. 1987. Some linearly induced Morse-Smale systems, the QR algorithm and the Toda lattice. Contemporary Mathematics, 64, 181–194. 118CrossRefGoogle Scholar
Soshnikov, A. 1999. Universality at the edge of the spectrum in Wigner random matrices. Communications in Mathematical Physics, 207(3), 697–733. 125CrossRefGoogle Scholar
Stam, AJ. 1982. Limit theorems for uniform distributions on spheres in high-dimensional Euclidean spaces. Journal of Applied Probability, 19(1), 221–228. 125CrossRefGoogle Scholar
Symes, WW. 1980. Hamiltonian group actions and integrable systems. Physica 1D, 4(1), 339–374. 10, 75Google Scholar
Symes, WW. 1982. The QR algorithm and scattering for the finite nonperiodic Toda lattice. Physica D: Nonlinear Phenomena, 4(2), 275–280. 10, 66, 75, 77CrossRefGoogle Scholar
Tao, T, and Vu, V. 2010. Random matrices: Universality of local Eigenvalue statistics up to the edge. Communications in Mathematical Physics, 298(2), 549–572. 125CrossRefGoogle Scholar
Tomei, C. 1984. The topology of isospectral manifolds of tridiagonal matrices. Duke Mathematical Journal, 51, 981–996. 112, 121CrossRefGoogle Scholar
Torres, D, and Tomei, C. 2013. An Atlas adapted to the Toda flow. International Mathematics Research Notices, 2013(16), 13867–13908. 104, 113, 118Google Scholar
Tracy, CA, and Widom, H. 1994. Level-spacing distributions and the Airy kernel. Communications in Mathematical Physics, 159, 151–174. 125CrossRefGoogle Scholar
Venakides, S, Deift, P, and Oba, R. 1991. The Toda shock problem. Communications on Pure and Applied Mathematics, 44(8–9), 1171–1242. 13CrossRefGoogle Scholar
Warner, FW. 1983. Foundations of Differentiable Manifolds and Lie Groups. Graduate Texts in Mathematics, vol. 94. Springer. 16, 21CrossRefGoogle Scholar
Witte, NS, Bornemann, F, and Forrester, PJ. 2013. Joint distribution of the first and second Eigenvalues at the soft edge of unitary ensembles. Nonlinearity, 26(6), 1799–1822. 126, 133, 134CrossRefGoogle Scholar
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