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 Riemann z- 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)