Crossing probabilities on same-spin clusters in the two-dimensional Ising model

TL;DR

This paper extends theoretical methods to derive a differential equation for crossing probabilities in the 2D Ising model, providing precise measurements and confirming the model's predictions with high accuracy.

Contribution

It introduces a sixth-order differential equation for crossing probabilities and improves measurement precision, validating the theoretical framework for the 2D Ising model.

Findings

Derived a 6th-order differential equation for crossing probability pih.

Achieved more precise measurements with 95% confidence interval of 4x10^{-4}.

Confirmed the theoretical solution closely matches empirical data.

Abstract

Probabilities of crossing on same-spin clusters, seen as order parameters, have been introduced recently for the critical 2d Ising model by Langlands, Lewis and Saint-Aubin. We extend Cardy's ideas, introduced for percolation, to obtain an ordinary differential equation of order 6 for the horizontal crossing probability pih. Due to the identity pih(r)+pih(1/r)=1, the function pih must lie in a 3-dimensional subspace. New measurements of pih are made for 40 values of the aspect ratio r (r in [0.1443,6.928]). These data are more precise than those obtained by Langlands et al as the 95%-confidence interval is brought to 4x10^{-4}. A 3-parameter fit using these new data determines the solution of the differential equation. The largest gap between this solution and the 40 data is smaller than 4x10^{-4}. The probability pihv of simultaneous horizontal and vertical crossings is also treated.