Rational layers in the low temperature expansion of the three-dimensional Ising model
Ran Huang
Abstract
A low-temperature expansion orders Ising droplets by wall area, but droplets with different transverse spans can occur at the same order. We instead organize the zero-field simple-cubic model by the excess above the minimum wall area compatible with two prescribed spans. We prove that every fixed excess layer, including all logarithmic cover corrections and Euler inversion, has a rational bivariate generating function with denominator factors restricted to 1 − x, 1 − y and 1 − x − y. The proof compresses repeated coordinate cuts into bounded decorations and assigns each unbounded interval to one monotone chain. A finite controller then enforces induced contacts and arbitrary cover-color reuse. We explicitly assemble the complete layers of excess zero through three over all widths, attributing their known geometric constituents to earlier polyomino enumeration. Exact lattice checks corroborate the formulas. A certified low-temperature comparison shows a modest hybrid accuracy gain at one sampling point and higher evaluation cost. The result establishes an exact layer structure; summation over unbounded excess remains a separate problem.