Aspects of Coulomb Gases

on YouTube ↗59m

" summary

Defines Coulomb kernel for general d, highlights the logarithmic kernel in 2D as singular and long-range, and shows equilibrium measure as the unique minimizer of EV with Euler–Lagrange condition (potential constant on support). Connects classical models (Ginibre, GUE, CUE, product-of-two-Ginibre → Cauchy heavy tails) and explains why beta=2 determinantal case yields exact formulas. Uses potential theory, Laplacian/fractional Laplacian, large deviations and Laplace method to get global law and shows Gaussian free field–type fluctuations (H^{-1} covariance) plus Langevin dynamics and McKean–Vlasov limits for evolution.

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" chapters (12)

1. 0:00Start
2. 0:47Coulomb kernel in mathematical physics
3. 5:34Coulomb potential and Coulomb energy
4. 9:23External field equilibrium measure
5. 13:43Convexity and Bochner positivity
6. 17:07Examples of equilibrium measures
7. 21:56Coulomb gas or one-component plasma
8. 25:12Examples: exactly solvable RMT and Coulomb gases
9. 33:23Laplace method point of view Laplace point of view
10. 40:47Asymptotic analysis of fluctuations
11. 44:43Heuristics beyond Coulomb case?
12. 47:02Langevin dynamics

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