### Statistical mechanics of Monte Carlo sampling and the sign problem

Jorge Kurchan, Paris

Monte Carlo sampling of any system may be analyzed in terms of an
associated glass model -- a variant of the Random Energy Model -- with,
whenever there is a sign problem, complex fields. This model has three
types of phases (frozen, chaotic and liquid), as is characteristic of
glass models with complex parameters. Only the liquid one
yields the correct answers for the original problem, and the task is to
design the simulation to stay inside it.
The statistical convergence of the sampling to the correct expectation
values may be studied in these terms, yielding a general lower bound of
the computer time as a function of the free energy difference between
the true system, and a reference one.