Critical Temperature(s) of Sierpinski Carpet(s)

TL;DR

This paper improves an algorithm for calculating the critical temperature of the Ising model on Sierpinski carpets, achieving higher generations and more accurate estimates through optimized transfer matrices and computational methods.

Contribution

It introduces a real-valued transfer matrix reformulation that significantly reduces computational complexity, enabling analysis of higher-generation Sierpinski carpets and more precise critical temperature estimates.

Findings

Achieved critical temperature estimate for SC(3,1) with high precision.

Extended analysis to other Sierpinski carpets and reported their critical temperatures.

Enabled analysis up to generation 10 for SC(3,1) using optimized algorithms.

Abstract

We present a key algorithmic improvement to the generalized combinatorial Feynman--Vdovichenko method for calculating the critical temperature of the Ising model on Sierpinski carpets $SC_k(a,b)$, originally introduced in arxiv:1505.02699. By reformulating the method in terms of purely real-valued transfer matrices, we substantially reduce their dimension. This optimization, together with modern computational resources, enables us to reach generation $k=10$ for the canonical $SC_k(3,1)$ carpet. Extrapolation from these data yields the most accurate estimate to date of the critical temperature $T_c^{(3,1)} = 1.4782927(26)$. We further extend the analysis to additional members of the $SC_k(a,b)$ family and report their corresponding critical temperatures.