  By developing a method to represent the Renyi entropies via a replica-trick
on classical statistical mechanical systems, we introduce a procedure to
calculate the Renyi Mutual Information in any Monte Carlo simulation. Through
simulations on several classical models, we demonstrate that the Renyi Mutual
Information can detect finite-temperature critical points, and even identify
their universality class, without knowledge of an order parameter or other
thermodynamic estimators. Remarkably, in addition to critical points mediated
by symmetry breaking, the Renyi Mutual Information is able to detect
topological vortex-unbinding transitions, as we explicitly demonstrate on
simulations of the XY model.
