  Turbulent flows in a thin layer can develop an inverse energy cascade leading
to spectral condensation of energy when the layer height is smaller than a
certain threshold. These spectral condensates take the form of large-scale
vortices in physical space. Recently, evidence for bistability was found in
this system close to the critical height: depending on the initial conditions,
the flow is either in a condensate state with most of the energy in the
two-dimensional (2-D) large-scale modes, or in a three-dimensional (3-D) flow
state with most of the energy in the small-scale modes. This bistable regime is
characterised by the statistical properties of random and rare transitions
between these two locally stable states. Here, we examine these statistical
properties in thin-layer turbulent flows, where the energy is injected by
either stochastic or deterministic forcing. To this end, by using a large
number of direct numerical simulations (DNS), we measure the decay time
$\tau_d$ of the 2-D condensate to 3-D flow state and the build-up time $\tau_b$
of the 2-D condensate. We show that both of these times $\tau_d,\tau_b$ follow
an exponential distribution with mean values increasing faster than
exponentially as the layer height approaches the threshold. We further show
that the dynamics of large-scale kinetic energy may be modeled by a stochastic
Langevin equation. From time-series analysis of DNS data, we determine the
effective potential that shows two minima corresponding to the 2-D and 3-D
states when the layer height is close to the threshold.
