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Random Matrices, Orthogonal Polynomials and Integrable Systems

Random Matrices, Orthogonal Polynomials and Integrable Systems. CRM-ISM colloquium Friday, Oct. 1, 2004. John Harnad. I.1. Introduction. Some history. 1950’s-60’s: (Wigner, Dyson, Mehta) Mainly the statistical theory of spectra of large nuclei.

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Random Matrices, Orthogonal Polynomials and Integrable Systems

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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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