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Analyzing dynamics and average case complexity in the spherical Sherrington-Kirkpatrick model: a focus on extreme eigenvectors
Published 8 Jan 2024 in math.PR, cs.NA, math-ph, math.MP, and math.NA | (2401.03668v1)
Abstract: We explore Langevin dynamics in the spherical Sherrington-Kirkpatrick model, delving into the asymptotic energy limit. Our approach involves integro-differential equations, incorporating the Crisanti-Horner-Sommers-Cugliandolo-Kurchan equation from spin glass literature, to analyze the system's size and its temperature-dependent phase transition. Additionally, we conduct an average case complexity analysis, establishing hitting time bounds for the bottom eigenvector of a Wigner matrix. Our investigation also includes the power iteration algorithm, examining its average case complexity in identifying the top eigenvector overlap, with comprehensive complexity bounds.
- M. Abramowitz and I. A. Stegun. Handbook of mathematical functions with formulas, graphs, and mathematical tables, volume 55. US Government printing office, 1948.
- Aging of spherical spin glasses. Probability theory and related fields, 120(1):1–67, 2001.
- A. Auffinger and W.-K. Chen. On the energy landscape of spherical spin glasses. Advances in Mathematics, 330:553–588, 2018.
- G. Ben-Arous. Aging and spin-glass dynamics. arXiv preprint math/0304364, 2003.
- Numerical analysis, brooks, 1997.
- S. Chatterjee. Estimation in spin glasses: A first step. Annals of Statistics, 35(5):1931 – 1946, 2007.
- K. L. Chung. A course in probability theory. Academic press, 2001.
- A. Crisanti and H.-J. Sommers. The spherical p-spin interaction spin glass model: the statics. Zeitschrift für Physik B Condensed Matter, 87(3):341–354, 1992.
- L. F. Cugliandolo and J. Kurchan. Analytical solution of the off-equilibrium dynamics of a long-range spin-glass model. Physical Review Letters, 71(1):173, 1993.
- L. F. Cugliandolo and J. Kurchan. On the out-of-equilibrium relaxation of the sherrington-kirkpatrick model. Journal of Physics A: Mathematical and General, 27(17):5749, 1994.
- P. Deift and T. Trogdon. Universality for eigenvalue algorithms on sample covariance matrices. SIAM Journal on Numerical Analysis, 55(6):2835–2862, 2017.
- A. Dembo and R. Gheissari. Diffusions interacting through a random matrix: universality via stochastic taylor expansion. Probability Theory and Related Fields, 180:1057–1097, 2021.
- A. Dembo and E. Subag. Dynamics for spherical spin glasses: disorder dependent initial conditions. Journal of Statistical Physics, 181:465–514, 2020.
- Message-passing algorithms for compressed sensing. Proceedings of the National Academy of Sciences, 106(45):18914–18919, 2009.
- R. Durrett. Probability: theory and examples, volume 49. Cambridge university press, 2019.
- Theory of spin glasses. Journal of Physics F: Metal Physics, 5(5):965, 1975.
- H.-O. Georgii. Gibbs measures and phase transitions, volume 9. Walter de Gruyter, 2011.
- The random geometry of equilibrium phases. In Phase transitions and critical phenomena, volume 18, pages 1–142. Elsevier, 2001.
- E. Kostlan. Complexity theory of numerical linear algebra. Journal of Computational and Applied Mathematics, 22(2-3):219–230, 1988.
- E. Kostlan. Statistical complexity of dominant eigenvector calculation. Journal of Complexity, 7(4):371–379, 1991.
- B. Landon. Free energy fluctuations of the two-spin spherical sk model at critical temperature. Journal of Mathematical Physics, 63(3):033301, 2022.
- J. O. Lee and J. Yin. A necessary and sufficient condition for edge universality of wigner matrices. Duke Mathematical Journal, 163(1):117–173, 2014.
- High-dimensional asymptotics of Langevin dynamics in spiked matrix models. Information and Inference: A Journal of the IMA, 12(4):2720–2752, 10 2023.
- B. Øksendal. Stochastic differential equations. In Stochastic differential equations, pages 65–84. Springer, 2003.
- D. Sherrington and S. Kirkpatrick. Solvable model of a spin-glass. Physical review letters, 35(26):1792, 1975.
- S. Smale. Complexity theory and numerical analysis. Acta numerica, 6:523–551, 1997.
- Spin glasses and complexity, volume 4. Princeton University Press, 2013.
- T. Tao and V. Vu. Random matrices: universal properties of eigenvectors. Random Matrices: Theory and Applications, 1(01):1150001, 2012.
- C. A. Tracy and H. Widom. Level-spacing distributions and the airy kernel. Communications in Mathematical Physics, 159:151–174, 1994.
- C. A. Tracy and H. Widom. On orthogonal and symplectic matrix ensembles. Communications in Mathematical Physics, 177:727–754, 1996.
- Slow dynamics and aging in spin glasses. In Complex Behaviour of Glassy Systems: Proceedings of the XIV Sitges Conference Sitges, Barcelona, Spain, 10–14 June 1996, pages 184–219. Springer, 2007.
- L. Zdeborová and F. Krzakala. Statistical physics of inference: Thresholds and algorithms. Advances in Physics, 65(5):453–552, 2016.
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