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Two-Dimensional Electrostatics and Universality in Random Matrix Theory -
What do parked cars, perched birds, and charged particles have in common? New arrivals to each ecosystem experience pressure from their neighbors not to get too close to them while also trying not to get out too far from the prime real estate. Suppose you observed yet another situation in which this neighborly repulsion seems to occur. Then you'd have a good candidate for yet another phenomenon which behaves like the eigenvalues of random matrices. I will generate some random matrices in hopes of demonstrating the universality of Wigner's Semicircle Law (akin to the universality of the Central Limit Theorem): The asymptotic behavior of random matrices (as the size and therefore number of eigenvalues increase without bound) often does not depend on the distributions from which the matrix entries are sampled. Under the appropriate assumptions about how charged particles repel one another, the electrostatic model of log-gas particles in the plane can mimic the eigenvalue statistics of certain classes of random matrices. The greater versatility in the log-gas model then promises new universality results.
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