Jon Machta

Graphical representations and cluster algorithms for critical points with fields

A two-replica graphical representation and associated cluster algorithm is described that is applicable to ferromagnetic Ising systems with arbitrary fields. Critical points are associated with the percolation threshold of the graphical representation. Results from numerical simulations of the Ising model in a staggered field are presented. The dynamic exponent for the algorithm is measured to be less than 0.5.

PARALLEL INVADED CLUSTER ALGORITHM FOR THE ISING MODEL

A parallel version of the invaded cluster algorithm is described. Results from large scale (up to 40962 and 5123) simulations of the Ising model are reported. No evidence of critical slowing down is found for the three-dimensional Ising model. The magnetic exponent is estimated to be 2.482±0.001(β/ν=0.518±0.001) for the three-dimensional Ising model.

Introduction to Focus Issue on “Randomness, Structure, and Causality: Measures of Complexity from Theory to Applications”

We introduce the contributions to this Focus Issue and describe their origin in a recent Santa Fe Institute workshop.

Yucesoy, Katzgraber, and Machta reply:.

criticize the conclusions of our Letter [2]. They argue that considering the Edwards-Anderson (EA) and SherringtonKirkpatrick (SK) models at the same temperature T is inappropriate and propose an interpretation based on replica symmetry breaking (RSB). In our Letter we compare the SK and EA models at the same low reduced temperature T ≈ 0.4Tc. Billoire et al. compare them at different T such that P (q = 0) is nearly the same. They also consider the quantity ∆(q0, κ) [2], which measures the probability with respect to the distribution of couplings J that PJ(q) exceeds κ in the range |q| < q0. In the low-T phase ∆ → 0 if a two-state picture holds, while ∆ → 1 if RSB holds. Considering the same T for both models was not essential to our argument; however, we think it is important to study both models at the lowest temperature possible to understand the low-T phase. For the EA model, we simulated systems up to size L = 12 at T = 0.423, whereas Billoire et al. simulated sizes up to L = 32 but at T = 0.703. We find ∆ leveling off as a function of L at low T (see Fig. 5, Ref. [2]); they find it increasing as a function of L at higher T (Fig. 1 inset, Ref. [1]). It is not clear which trade-off in L vs T better reflects the infinite-volume behavior. However, P (q) for L = 12 and T = 0.42 is closer to a δ function at qEA, which is the infinite-volume behavior: P (qEA) divided by the width at half maximum of the qEA peak equals 29.1 for L = 12 at T = 0.423, and 18.4 for L = 32 and T = 0.703 [3]. We also note that the increase in ∆ seen in Fig. 1 (inset) of Ref. [1] is most pronounced forL = 32. However, this point appears to be anomalous and P (q) from the same simulations [3] shows a similar anomaly, which may reflect large statistical errors or incomplete equilibration. Finally, Ref. [1] studies bimodal disorder, which converges slower [4] than Gaussian. The theory in the Comment attributes the plateau in ∆ for our EA data to a combination of a small value of I(q0) = ∫ |q|<q0 P (q)dq and the slow growth in L of P (qEA). It predicts that ∆ for the EA model will grow to unity extremely slowly in L. Our Fig. 6 shows that even after factoring out the slower growth in P (qEA) for the EA model compared to the SK model, we still find a qualitative difference between the two. The proposed RSB scaling theory [1] asserts that ∆(q0, κ) ∼ [P (qEA)/κ]0 − 1. This can be simplified when I(q0) is small. Noting that I(q0) ≈ q0P (0) one obtains ∆(q0, κ) ≈ q0P (0) log[P (qEA)/κ]. The predicted linear dependence of ∆ on q0 is consistent with our data and is neither surprising nor a strong test of the theory. The fact that data from different sizes lie on similar curves agrees with the plateau in our data but does not demonstrate that ∆ is actually growing slowly with L for fixed q0 and κ. To test the validity of the proposed theory [1], we compared our data for the EA model at several T holding P (qEA)/κ and I(q0) fixed. The proposed theory predicts that if these variables are fixed, ∆ should remain constant. Figure 1 shows ∆(q0, κ) vs T for L = 10, 12. For each T and L, both q0 and κ are adjusted so that I(q0) ≈ 0.067 and P (qEA)/κ = 3 (q0 ranges from 0.16 to 0.56 and κ from 0.5 to 2.6 as T decreases from 0.7 to 0.2).

Parallel complexity of random Boolean circuits

Random instances of feedforward Boolean circuits are studied both analytically and numerically. Evaluating these circuits is known to be a P-complete problem and thus, in the worst case, believed to be impossible to perform, even given a massively parallel computer, in a time much less than the depth of the circuit. Nonetheless, it is found that, for some ensembles of random circuits, saturation to a fixed truth value occurs rapidly so that evaluation of the circuit can be accomplished in much less parallel time than the depth of the circuit. For other ensembles saturation does not occur and circuit evaluation is apparently hard. In particular, for some random circuits composed of connectives with five or more inputs, the number of true outputs at each level is a chaotic sequence. Finally, while the average case complexity depends on the choice of ensemble, it is shown that for all ensembles it is possible to simultaneously construct a typical circuit together with its solution in polylogarithmic parallel time.