Dynamical invariance for random matrices

We consider a general Langevin dynamics for the one-dimensional N-particle Coulomb gas with confining potential $V$ at temperature $β$. These dynamics describe for $β=2$ 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 Schrödinger-Virasoro algebra.

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