  Fyodorov, Hiary & Keating established an intriguing connection between the
maxima of log-correlated processes and the ones of the Riemann zeta function on
a short interval of the critical line. In particular, they suggest that the
analogue of the free energy of the Riemann zeta function is identical to the
one of the Random Energy Model in spin glasses. In this paper, the connection
between spin glasses and the Riemann zeta function is explored further. We
study a random model of the Riemann zeta function and show that its two-overlap
distribution corresponds to the one of a one-step replica symmetry breaking
(1-RSB) spin glass. This provides evidence that the local maxima of the zeta
function are strongly clustered.
