Validity of hyperscaling for thed=3Ising model

The availability of extended series expansions for ${\chi }_0^{\mathrm{}\left(2\right)\mathrm{}}$=${\partial }^2$χ/∂$H^2$, accurate Monte Carlo estimates of the critical temperature for the sc spin-(1/2) Ising model, and refined methods of series analysis allow the hyperscaling relation dν=2Δ-γ to be severely tested. From analysis of ${\chi }_0^{\mathrm{}\left(2\right)\mathrm{}}$ and ${\chi }_0^{\mathrm{}\left(2\right)\mathrm{}}$/${\chi }_0^2$ we estimate γ=1.240±0.006 and Δ=1.565±0.004. The estimate of Δ is in complete agreement with the prediction of hyperscaling using both the best extant series and renormalization-group estimates of the exponents γ and ν. With the assumption of hyperscaling, our analysis gives ν=0.630±0.002.