TY - JOUR

T1 - Extrapolating Monte Carlo simulations to infinite volume

T2 - Finite-size scaling at ξ/L 1

AU - Caracciolo, Sergio

AU - Edwards, Robert G.

AU - Ferreira, Sabino José

AU - Pelissetto, Andrea

AU - Sokal, Alan D.

N1 - Funding Information:
ment with theory \[5\]I. n practice we obtained ~oo to an accuracy of about 0.2% (resp. 0.7%, 1.1%, 1.6%) at {oo ~ 102 (resp. 103, 104, 10s). We also carried out a "simulated Monte Carlo" experiment for the O(N) ~-model at N = oo, by generating data from the exact finite-volume solution plus random noise of 0.1% for L = 64, 96,128, 0.2% for L = 192,256 and 0.5% for L = 384,512 \[which is the order of magnitude we attain in practice for 0(3)\]. We considered 35 values of /3 in the range 20 < ~oo < 106 . We used ~min -~" 20 and Lmin = 64 (in fact much smaller values could have been used, as corrections to scaling are here very small) and a ninth-order fit; for two different data sets we get X 2 = 114 (resp. 118) with 166 DF. In practice we obtain ~ with an accuracy of 0.6% (resp. 1.2%, 2%, 3%) at ~oo ~ 10 a (resp. 104, 105, 106). Here we can also compare the extrapolated values ~oxt'(/3) with the exact values ~act(/3). Defining T~ = EO\[~oo e~tr (/3)-~ga~t(~3)\]2/~z(~), we find for the two data sets 7¢ = 17.19 (resp. 25.81) with 35 DF. Only 6 (resp. 9) points differ from the exact value more than one standard deviation, and none by more than two. Details on all of these models will be reported separately \[2,4\]. The method is easily generalized to a model controlled by an RG fixed point having k relevant operators. It suffices to choose k - 1 dimensionless ratios of long-distance observables, call them R = (Rt,..-,Rk-1); then the function Fo will depend parametrically on R(/3, L). In practice one can divide R-space into "slices" within which Fo is empirically constant within error bars, and perform the fit (3) within each slice. We have used this approach to study the mixed isovec-tor/isotensor ~r-model, taking R to be the ratio of isovector to isotensor correlation length \[3,4\]. The method can also be applied to extrapolate the exponential correlation length (inverse mass gap). For this purpose one must work in a system of size L a-a x T with T > ~v(/3, L) (cf. \[8\]). We wish to thank Martin Hasenbusch and especially Jae-Kwon Kirn for sharing their data with us, and for challenging us to push to ever larger values of (/L. This research was supported by CNR, 1NFN, CNPq, FAPEMIG, DOE contracts DE-FG05-85ER250000 and DE-FG05-92ER40742, NSF grant DMS-9200719, and NATO CRG 910251.
Copyright:
Copyright 2015 Elsevier B.V., All rights reserved.

PY - 1995

Y1 - 1995

N2 - We present a simple and powerful method for extrapolating finite-volume Monte Carlo data to infinite volume, based on finite-size-scaling theory. We discuss carefully its systematic and statistical errors, and we illustrate it using three examples: the two-dimensional three-state Potts antiferromagnet on the square lattice, and the two-dimensional O(3) and O() σ models. In favorable cases it is possible to obtain reliable extrapolations (errors of a few percent) even when the correlation length is 1000 times larger than the lattice.

AB - We present a simple and powerful method for extrapolating finite-volume Monte Carlo data to infinite volume, based on finite-size-scaling theory. We discuss carefully its systematic and statistical errors, and we illustrate it using three examples: the two-dimensional three-state Potts antiferromagnet on the square lattice, and the two-dimensional O(3) and O() σ models. In favorable cases it is possible to obtain reliable extrapolations (errors of a few percent) even when the correlation length is 1000 times larger than the lattice.

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U2 - 10.1103/PhysRevLett.74.2969

DO - 10.1103/PhysRevLett.74.2969

M3 - Article

AN - SCOPUS:3543015244

VL - 74

SP - 2969

EP - 2972

JO - Physical Review Letters

JF - Physical Review Letters

SN - 0031-9007

IS - 15

ER -