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 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)

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)

I.3. Some pictures- Wigner semicircle law (GUE)- GUE (and Riemann ) pair correlations - GUE (and Riemann ) 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)

Wigner semicircle law (GUE)

GUE (and Riemann zeros) pair correlations (Montgomery-Dyson)

Comparison of pair correlations of GUE

with zeros of Riemann function

GUE (and Riemann 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 2D Young tableaux (Cohn-Lars-Prop rotated)

2D random partition (dimer.cardioid: Okounkov)

Random 2D partitions (cardioid: Okounkov)

Other growth processes: diffusion limited aggregation

Laplacian growth:Viscous fingering in a Hele-Shaw cell

(click to animate)

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