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Advancements in Random Matrices and Integrable Systems: A Comprehensive Exploration

Delve into the historical influences and modern developments of random matrices, orthogonal polynomials, and integrable systems. Discover connections to quantum gravity, graphical enumeration, and rigorous asymptotics. Explore newer connections such as discrete orthogonal polynomials, determinantal growth processes, and multi-matrix models. Gain insights into various distributions and growth processes related to random sequences and partitions. Visualize phenomena like Wigner semicircle law, GUE correlations, Dyson processes, and random hexagon tilings. This colloquium offers a deep dive into the fascinating world of random matrix theory and integrable systems.

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Advancements in Random Matrices and Integrable Systems: A Comprehensive Exploration

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  1. Random Matrices, Orthogonal Polynomials and Integrable Systems CRM-ISM colloquium Friday, Oct. 1, 2004 John Harnad

  2. I.1. Introduction. Some history • 1950’s-60’s: (Wigner, Dyson, Mehta) Mainly the statistical theory of spectra of large nuclei. • Early 1990’s: Applications to 2D quantum gravity (Douglas, Moore) and graphical enumeration (Itzykson, Zuber, Zinn-Justin); heuristic large N asymptotics, “universality” • Late 1990’s - present: Rigorous large N asymptotics - Proofs of “universality” (Its- Bleher, Deift et al) - Riemann-Hilbert methods; integrable systems - Largest eigenvalue distributions (Tracy-Widom) - Relations to random sequences, partitions, words (Deift, Baik, Johansson, Tracy, Widom)

  3. I.2. Newer connections and developments • Discrete orthogonal polynomials ensembles, relations to “dimer” models ( Reshetikhin-Okounkov-Borodin) • Relations to other “determinantal” growth processes (“Polynuclear growth”: Prahofer-Spohn, Johansson) • Large N limits --> dispersionless limit of integrable systems (Normal and complex matrix models) - Relations to free boundary value problems in 2D- viscous fluid dynamics (Wiegmann-Zabrodin-Mineev) • Multi-matrix models, biorthogonal polynomials, Dyson processes (Eynard- Bertola-JH; Adler-van Moerbeke; Tracy-Widom)

  4. I.3. Some pictures • Wigner semicircle law (GUE) • GUE (and Riemann z) pair correlations • GUE (and Riemann z) spacing distributions • Edge spacing distribution (Tracy-Widom) • Dyson processes (random walks of eigenvalues) • Random hexagon tilings (Cohn-Larson-Prop) • Random 2D partitions (Cohen-Lars-Prop rotated) • Random 2D partitions/dimers (cardioid bound: Okounkov) • Polynuclear growth processes (Prähofer and Spohn) • Other growth processes: diffusion limited aggregation • Laplacian growth (2D viscous fluid interfaces)

  5. Wigner semicircle law (GUE)

  6. GUE (and Riemann z zeros) pair correlations (Montgomery-Dyson)

  7. Comparison of pair correlations of GUE with zeros of Riemannz- function

  8. GUE (and Riemann z zeros) spacing distributions (PV: Jimbo-Miwa)

  9. GUE edge spacing distributions (PII: Tracy-Widom)

  10. Dyson processes: eigenvalues of a hermitian matrix undergoing a Gaussian random walk.

  11. Polynuclear growth processes (Prähofer and Spohn)

  12. Random hexagon aztec tilings (Cohen-Lars-Prop)

  13. Random 2D Young tableaux (Cohn-Lars-Prop rotated)

  14. 2D random partition (dimer.cardioid: Okounkov)

  15. Random 2D partitions (cardioid: Okounkov)

  16. Other growth processes: diffusion limited aggregation

  17. Laplacian growth:Viscous fingering in a Hele-Shaw cell (click to animate)

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