Statistical mechanics of Monte Carlo sampling and the sign problem

¹ Laboratoire de Physique Statistique, Ecole Normale Superieure, associé au CNRS et aux Universités Paris VI et VII - 24 rue Lhomond, 75005 Paris, France, EU
² PMMH, ESPCI - 10 rue Vauquelin, CNRS UMR 7636, 75005 Paris, France, EU

Received: 5 October 2010
Accepted: 10 November 2010

Abstract

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 (liquid, frozen and “chaotic”), 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 for the computer time as a function of the free energy difference between the true system, and a reference one. In this way, importance sampling strategies may be optimized.