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5
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84927466949
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It is necessary to choose a definition of ξ that makes sense in a fully finite system. We use the second-moment correlation length defined by Eqs. (4.11)–(4.13) of Ref. [c6].
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7
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84927464047
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This form of finite-size scaling assumes hyperscaling, and thus is expected to hold only below the upper critical dimension of the model. See, e.g., Ref. [c1], Chap. I, Sec. 2.7.
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8
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84927489219
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Our method has many features in common with those of Lüscher, Weisz, and Wolff [c9] and Kim [c10]. In particular, all these methods share the property of working only with observable quantities ( ξ,O, and L) and not with bare quantities (β). Therefore, they rely on “scaling” and not on “asymptotic scaling;” they differ from other FSS-based methods such as phenomenological renormalization.
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In principle, ξ and O should be generated from a joint Gaussian with the correct covariance. We ignored this subtlety and simply generated independent fluctuations on ξ and O.
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17
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84927463753
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Errors of types (i) and (ii) can be exactly computed in terms of fξ and fO. For ξ we have Δξ∞/ξ∞=Kξ(ξ∞/L)ΔξL/ξL, where ΔξL ( Δξ∞) is the standard deviation on the raw (extrapolated) value, and Kξ(z)=fξ(z)/[ fξ(z)+zfξ′(z)]. For a deviation, see Ref. [c4].
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18
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84927506162
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The extrapolated estimates from different (β,L) are correlated. In setting the error bars we have kept account of the full covariance matrix between extrapolations at the same β but different L, but we have ignored correlations between extrapolations at different values of β.
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84927488332
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This variance-time product tends to a constant as the CPU time tends to infinity. However, if the CPU time used is too small, then the variance-time product can be significantly larger than its asymptotic value, due to nonlinear cross terms between error sources (i) and (ii).
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84927486681
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The exponent p, which does not depend on s, can be computed in terms of Fξ. Define an exponent h by assuming that ΔξL/ξL≈ξ∞zH(ξ∞/L)/Niter∼ξ∞z+h/Niter, where ΔξL is the error on ξL for a run of length Niter. If ξL/L→x*<∞, when ξ∞/L→∞ (this is the case for models with critical exponent η>0) and R≡1+x*Fξ′(x*)/s>1, then p=-d+2 logR/logs+2h. If R=1, logarithmic terms appear and Gξ(z)∼z-d+2h(logz)q, with q>0. If instead ξL/L→∞, when ξ∞/L→∞ (this is the case for asymptotically free theories), then Gξ(z)∼z-d+2h(logz)2.
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In our three examples, the total error (i) + (ii) + (iii) is never more than twice the error (i) + (ii), and is usually much less.
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