Resetting Dyson Brownian motion

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) -- in short the β-RDBM. For β= 1, 2, 4, the positions of the particles in the RDBM can be interpreted as the eigenvalues of a random matrix ensemble where the entries of an N x 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 observables for large N: the average density profile of the gas, the extreme value statistics, the spacing between two consecutive particles and the full counting statistics. We show that a nonzero resetting rate r>0 drastically changes the nature of the fluctuations in the stationary state: while the log-gas 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 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.