  Recently, we introduced the active Dyson Brownian motion model (DBM), in
which $N$ run-and-tumble particles interact via a logarithmic repulsive
potential in the presence of a harmonic well. We found that in a broad range of
parameters the density of particles converges at large $N$ to the Wigner
semi-circle law, as in the passive case. In this paper, we provide an
analytical support for this numerical observation, by studying the fluctuations
of the positions of the particles in the nonequilibrium stationary state of the
active DBM, in the regime of weak noise and large persistence time. In this
limit, we obtain an analytical expression for the covariance between the
particle positions for any $N$ from the exact inversion of the Hessian matrix
of the system. We show that, when the number of particles is large $N \gg 1$,
the covariance matrix takes scaling forms that we compute explicitly both in
the bulk and at the edge of the support of the semi-circle. In the bulk, the
covariance scales as $N^{-1}$, while at the edge, it scales as $N^{-2/3}$.
Remarkably, we find that these results can be transposed directly to an
equilibrium model, the overdamped Calogero-Moser model in the low temperature
limit, providing an analytical confirmation of the numerical results by
Agarwal, Kulkarni and Dhar. For this model, our method also allows us to obtain
the equilibrium two-time correlations and their dynamical scaling forms both in
the bulk and at the edge. Our predictions at the edge are reminiscent of a
recent result in the mathematics literature by Gorin and Kleptsyn on the
(passive) DBM. That result can be recovered by the present methods, and also,
as we show, using the stochastic Airy operator. Finally, our analytical
predictions are confirmed by precise numerical simulations, in a wide range of
parameters.
