Dynamic weighting in simulations of spin systems

We establish the invariance property of a dynamically weighted Monte Carlo process and apply the method to the simulation of spin glasses and Ising models. For the two-dimensional Edwards-Anderson model, we obtain an ergodicity time of O( L2.44), where L is the linear dimension of the lattice. The results suggest that dynamic weighting is a highly promising new tool for Monte Carlo simulation. @ 1999 Published by Elsevier Science B.V. PAC.? 02.7O.Lq; 05SO.+q; 75.10.Nr Kqywordst Dynamic weighting; king model; Invariance; Spin glass At low temperature spin glasses and other complex systems have many local energy minima. Conventional Monte Carlo simulations are ineffective for such systems due to the difficulty in crossing the steep energy barriers separating the minima. A major goal of current Monte Carlo research is to find ways to overcome this difficulty. Although the cluster algorithms [ 1,2] have greatly reduced the autocorrelation time at criticality for the second order phase transition, they are not as effective in the case of the first order phase transition. In this respect, simulated tempering [ 31, exchange Monte Carlo [ 14,15,7], multicanonical algorithm [ 4,5] and a related algorithm [ 161 have presented some substantial progress. In simulated tempering, temperature is treated as a dynamic variable taking values in a ladder of suitably chosen levels near the critical temperature. This method has been very successfully in the simulation of spin glasses [6,7] and random field Ising models [3]. However, it performed less well if many levels are needed to cover a large energy variation such as in simulating an Ising system at a temperature below the critical point. We have recently proposed dynamic weighting [ 81 as a new algorithm for general simulation and optimization tasks. This method relies on an additional dynamic variable, the importance weight, to help the system overcome steep energy barriers. A notable aspect of the proposed transition rule is that they do not necessarily satisfy detailed balance. Instead, it was proposed that the property of invariance with respect to importance weighting (IWIW) is used to guide the design of proper transition rules. In this Letter, we discuss the application of dynamic weighting to the simulation of Ising models and spin glasses. We outline the proof of the IWIW property for the transition rules, and present numerical evidence that the method has succeeded in achieving a great reduction in ergodicity times. Suppose our task is to simulate a system x according to a density f(x). We augment the system to be 0375-9601/99/$ see front matter @ 1999 Published by Elsevier Science B.V. All rights reserved. PII SO375-9601(99)00006-7 2% E Licmg. U? H. Won,q/Physics Letrers A 252 (1999) 257-262 (x. w), where IV is a dynamic variable representing the importance weight for X. The following transition rule, called a weighted transition, is an important ingredient of our algorithm: draw v from a proposal transition function T( x + y) (T(x ---f _v) take values from a domain where f(y) > 0) and compute the Metropolis-Hastings (91 ratio g,(x’, M”) = c Cgo(x, w)T(x --f x’) r = rfx,yf f(y) T(” .r^) f(s) T(x ---f y) . (1) The proposed transition is accepted with probability n = a(.~, M’; v) = tvr/(nr+B). If it is accepted, we set the new state to (,v, w’) with weight u“ = rvr/n; otherwise, the proposal is rejected and the state is set to (x, n’) with weight n:’ = w/( 1 -a). Here@ = @(x, rv) is a threshold function which can be used to control the ease of acceptance. A simulation run then produces a sequence of samples (Xi_ Wi), i = 1,2.. . . . The expectation Efp(x) for a state function p(x) can be estimated by the weighted average [c p(x;) wJ/[ c w;] over samples obtained after the process has reached equilibrium. We now verify that such a weighted transition satisfies IWIW, i.e.. it would transform correctly weighted densities to correctly weighted densities. We assume that the sample space X is finite, and that the initial weight is wo = 1. In this case, the weight space W is countable. Suppose gu(x, w) is a correctly weighted density, i.e., c go(x,w)w= cf(x) 1 (2) for some c, and gi (x’, NV’) is the density after one step transition from the density go(x, w), then our task is to show that gi (x’, I-V’) is also correctly weighted in the sense that c gl (x’, w’) w’ = c’f( x’) _ (3) 11.1 From the definition of the transition rule. we can write .c >Y x afx, w;x’)6 ( w’, wr a(r,w;x’f ) 4-)---;yyy_go(x.,w)T(X~+ y)(l --a(x,w;y))