Alessandro Codello — CP3-Origins Odense (DK) # 
Approximating the Ising model on fractal lattices of dimension below two # 

We construct an approximation to the free energy of the Ising model on fractal lattices of 
dimension smaller than two, in the case of zero external magnetic field. The result is obtained as the limit of 
the exact free energies of the Ising model on periodic approximations. The free energies are computed using a 
generalization of the combinatorial method of Feynman and Vodvickenko. 
<br/> 

As a first application, we compute estimates to the critical temperature for many different Sierpinski carpets 
and we compare the results with known Monte Carlo estimates. The results show that our method is capable of 
determining the critical temperature with, possibly, arbitrary accuracy and paves the way to determine \(T_c\) 
for any fractal of dimension below two.
<br/>
The singularity of the free energy is logarithmic at the critical point, thus \(\alpha = 0\), for any periodic 
approximation. We also compute the correlation length as a function of the temperature and extract the relative 
critical exponent, we find \(\nu=1\) for all periodic approximation, as expected from Universality.