Susceptibility amplitude ratios in the two-dimensional Potts model and percolation

Abstract

The high-temperature susceptibility of the $q$-state Potts model behaves as $Γ|T-T_c|^{-γ}$ as $T\to T_c+$, while for $T\to T_c-$ one may define both longitudinal and transverse susceptibilities, with the same power law but different amplitudes $Γ_L$ and $Γ_T$. We extend a previous analytic calculation of the universal ratio $Γ/Γ_L$ in two dimensions to the low-temperature ratio $Γ_T/Γ_L$, and test both predictions with Monte Carlo simulations for $q=3$ and 4. The data for $q=4$ are inconclusive owing to large corrections to scaling, while for $q=3$ they appear consistent with the prediction for $Γ_T/Γ_L$, but not with that for $Γ/Γ_L$. A simple extrapolation of our analytic results to $q\to1$ indicates a similar discrepancy with the corresponding measured quantities in percolation. We point out that stronger assumptions were made in the derivation of the ratio $Γ/Γ_L$, and our work suggests that these may be unjustified.