Resetting Dyson Brownian motion.

Abstract

In this paper, we introduce a new stochastic process of N interacting particles on the line that evolve via Dyson Brownian motion (DBM) with Dyson's index β>0 and undergo simultaneous resetting to their initial positions at a constant rate r. We call this process the resetting Dyson Brownian motion (RDBM) with a parameter β>0, in short the β-RDBM process. For β=1,2,4, the positions of the particles in the RDBM can be interpreted as the eigenvalues of a new random matrix ensemble where the entries of an N×N Gaussian matrix evolve as simultaneously resetting Brownian motions (with rate r) in the presence or absence of a harmonic trap. For r=0 and in the presence of a harmonic trap, this system reaches an equilibrium Gibbs-Boltzmann state of the so called Dyson log-gas. However, the stochastic resetting drives the system at long time to a nonequilibrium stationary state (NESS). We compute exactly the joint distribution of the positions of the particles in this NESS for all β>0 and calculate several macroscopic and microscopic observables in the large N limit. These include the average density profile of the gas, the extreme value statistics, the spacing between two consecutive particles and the full counting statistics, i.e., the distribution of the number of particles in an interval [-L,L]. We show that a nonzero resetting rate r>0 drastically changes the nature of the fluctuations in the stationary state: while the log-gas (r=0) is rather rigid, the β-RDBM in its NESS becomes fluffy, i.e., the fluctuations of different observables are of the same order as their mean. In the absence of a harmonic trap, our results for the β=2 RDBM can be related to nonintersecting Brownian motions (vicious walkers) in the presence of resetting. Our model demonstrates interesting effects arising from the interplay between the eigenvalue repulsion and the all-to-all attraction (generated by stochastic resetting) in an interacting particle system. Numerical simulations are in excellent agreement with our analytical results.