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

Abstract

The high-temperature susceptibility of the $q$-state Potts model behaves as $\Gamma |T-T_c{|}^{-\gamma }$ 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 ${\Gamma }_L$ and ${\Gamma }_T$. We extend a previous analytic calculation of the universal ratio $\Gamma /{\Gamma }_L$ in two dimensions to the low-temperature ratio ${\Gamma }_T/{\Gamma }_L$, 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 ${\Gamma }_T/{\Gamma }_L$, but not with that for $\Gamma /{\Gamma }_L$. A simple extrapolation of our analytic results to $q\to 1$ 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 $\Gamma /{\Gamma }_L$, and our work suggests that these may be unjustified.

17 pages, latex

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