  The expected return time of a fluctuating two-dimensional cluster or vacancy
to a given configuration is studied in thermodynamic equilibrium. We define a
family of bond-breaking models that preserve the number of particles. This
family includes edge diffusion and surface diffusion inside vacancies in the
limit of fast particle diffusion and slow attachment-detachment kinetics.
Within the frame of these bond-breaking models, the expected return time is
found to depend on the energies of the configurations and on the energies of
the excited states formed by removing a single particle from the cluster. High
and low temperature regimes are studied. We clarify the conditions under which
the return time is a non-monotonous function of temperature: a minimum is found
when the energy obtained by the average over the excited states of the
configuration weighted by their attachment probabilities is lower than the
energy averaged over all states. In addition, we show that the optimal
temperature at which the return time is minimum is shifted to a higher
temperature as compared to the temperature at which the equilibrium probability
is maximum. This shift is influenced by the average curvature of the cluster
edge, and is therefore larger for vacancies.
