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arXiv · 1603.09373

Dynamical invariance for random matrices

Jeremie Unterberger
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Abstract

We consider a general Langevin dynamics for the one-dimensional N-particle Coulomb gas with confining potential $V$ at temperature $\beta$. These dynamics describe for $\beta=2$ the time evolution of the eigenvalues of $N\times N$ random Hermitian matrices. The equilibrium partition function -- equal to the normalization constant of the Laughlin wave function in fractional quantum Hall effect -- is known to satisfy an infinite number of constraints called Virasoro or loop constraints. We introduce here a dynamical generating function on the space of random trajectories which satisfies a large class of constraints of geometric origin. We focus in this article on a subclass induced by the invariance under the Schr\"odinger-Virasoro algebra.

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Jeremie Unterberger. 2016-03-30. Dynamical invariance for random matrices. https://arxiv.org/abs/1603.09373

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