Parallel tempering, in physics and statistics, is a computer simulation method typically used to find the lowest energy state of a system of many interacting particles.
When the system is highly correlated, updates are rejected and the simulation is said to suffer from critical slowing down.
The overlap can be defined by the area of the histograms that falls over the same interval of energy values, normalized by the total number of samples.
Set up is important as there must be a practical histogram overlap to achieve a reasonable probability of lateral moves.
The parallel tempering method can be used as a super simulated annealing that does not need restart, since a system at high temperature can feed new local optimizers to a system at low temperature, allowing tunneling between metastable states and improving convergence to a global optimum.