Dynamical invariance for random matrices
Résumé
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×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.