Specific Heat of Ising Model with Holes: Mathematical Details Using Dimer Approaches

In this paper, we use the dimer method to obtain the free energy of Ising models consisting of repeated horizontal strips of width $m$ connected by sequences of vertical strings of length $n$ mutually separated by distance $N$, with $N$ arbitrary, to investigate the effects of connectivity and proximity on the specific heat. The decoration method is used to transform the strings of $n+1$ spins interacting with their nearest neighbors with coupling $J$ into a pair with coupling $\bar J$ between the two spins. The free energy per site is given as a single integral and some results for critical temperatures are derived.