A general limitation on Monte Carlo algorithms of the Metropolis type.

We prove that for any Monte Carlo algorithm of Metropolis type, the autocorrelation time of a suitable “energy”-like observable is bounded below by a multiple of the corresponding “specific heat”. This bound does not depend on whether the proposed moves are local or non-local; it depends only on the distance between the desired probability distribution π and the probability distribution π(0) for which the proposal matrix satisfies detailed balance. We show, with several examples, that this result is particularly powerful when applied to non-local algorithms. PACS number(s): 02.70.Lq, 02.50.Ga, 05.50.+q, 11.15.Ha ∗Address until August 31, 1994. Permanent address: Dipartimento di Fisica and INFN – Sezione di Pisa, Università degli Studi di Pisa, Pisa 56100, ITALIA. Internet: [email protected]; Bitnet: [email protected]; Hepnet/Decnet: 39198::PELISSETTO. Forty years ago, Metropolis et al. [1] introduced a general method for constructing dynamic Monte Carlo algorithms (= Markov chains [2]) that satisfy detailed balance for a specified probability distribution π. In this note we would like to point out a general limitation on all algorithms of Metropolis type. We prove that the autocorrelation time of a suitable “energy”-like observable is bounded below by a multiple of the corresponding “specific heat”. This bound does not depend on whether the proposed moves are local or non-local; it depends only on the distance between the desired probability distribution π and the probability distribution π for which the proposal matrix satisfies detailed balance. Let us begin by recalling the general Metropolis et al. [1] method, as slightly generalized by Hastings [3]. We use the notation of a discrete (finite or countably infinite) state space S, but the same considerations apply with minor modifications to a general measurable state space. Let P (0) = {p xy } be an arbitrary transition matrix on S. We call P (0) the proposal matrix , and use it to generate proposed moves x → y that will then be accepted or rejected with probabilities axy and 1−axy, respectively. If a proposed move is rejected, we make a “null transition” x → x. The transition matrix P = {pxy} of the full algorithm is thus