Random Matrices Orthogonal Polynomials and Integrable Systems CRMISM
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. • Early 1990’s: Applications to 2 D 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)
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 2 Dviscous fluid dynamics (Wiegmann-Zabrodin-Mineev) • Multi-matrix models, biorthogonal polynomials, Dyson processes (Eynard- Bertola-JH; Adler-van Moerbeke; Tracy-Widom)
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 2 D partitions (Cohen-Lars-Prop rotated) Random 2 D partitions/dimers (cardioid bound: Okounkov) Polynuclear growth processes (Prähofer and Spohn) Other growth processes: diffusion limited aggregation Laplacian growth (2 D viscous fluid interfaces)
Wigner semicircle law (GUE)
GUE (and Riemann z zeros) pair correlations (Montgomery-Dyson)
Comparison of pair correlations of GUE with zeros of Riemann z- function
GUE (and Riemann z zeros) spacing distributions (PV: Jimbo-Miwa)
GUE edge spacing distributions (PII: Tracy-Widom)
Dyson processes: eigenvalues of a hermitian matrix undergoing a Gaussian random walk.
Polynuclear growth processes (Prähofer and Spohn)
Random hexagon aztec tilings (Cohen-Lars-Prop)
Random 2 D Young tableaux (Cohn-Lars-Prop rotated)
2 D random partition (dimer. cardioid: Okounkov)
Random 2 D partitions (cardioid: Okounkov)
Other growth processes: diffusion limited aggregation
Laplacian growth: Viscous fingering in a Hele-Shaw cell (click to animate)
- Slides: 17