  A variant of the parallel tempering method is proposed in terms of a
stochastic switching process for the coupled dynamics of replica configuration
and temperature permutation. This formulation is shown to facilitate the
analysis of the convergence properties of parallel tempering by large deviation
theory, which indicates that the method should be operated in the infinite
swapping limit to maximize sampling efficiency. The effective equation for the
replica alone that arises in this infinite swapping limit simply involves
replacing the original potential by a mixture potential. The analysis of the
geometric properties of this potential offers a new perspective on the issues
of how to choose of temperature ladder, and why many temperatures should
typically be introduced to boost the sampling efficiency. It is also shown how
to simulate the effective equation in this many temperature regime using
multiscale integrators. Finally, similar ideas are also used to discuss
extensions of the infinite swapping limits to the technique of simulated
tempering.
