random matrices, orthogonal polynomials and integrable systems crm-ism colloquium friday, oct. 1,...
TRANSCRIPT
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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