  Parallel tempering, or replica exchange, is a popular method for simulating
complex systems. The idea is to run parallel simulations at different
temperatures, and at a given swap rate exchange configurations between the
parallel simulations. From the perspective of large deviations it is optimal to
let the swap rate tend to infinity and it is possible to construct a
corresponding simulation scheme, known as infinite swapping. In this paper we
propose a novel use of large deviations for empirical measures for a more
detailed analysis of the infinite swapping limit in the setting of continuous
time jump Markov processes. Using the large deviations rate function and
associated stochastic control problems we consider a diagnostic based on
temperature assignments, which can be easily computed during a simulation. We
show that the convergence of this diagnostic to its a priori known limit is a
necessary condition for the convergence of infinite swapping. The rate function
is also used to investigate the impact of asymmetries in the underlying
potential landscape, and where in the state space poor sampling is most likely
to occur.
