Theory of layered Ising models: Thermodynamics

We consider a two-dimensional Ising model whose vertical interaction energies $E_2\mathrm{}\left(j\right)\mathrm{}$ between row $j$ and row $j+1\mathrm{}$ are allowed to be arbitrary for $1\le j\le n$. This set of bonds is then repeated indefinitely to make up an infinite lattice. For any set of the $n$ energies $E_2\mathrm{}\left(j\right)\mathrm{}$ we show that the specific heat has a logarithmic divergence as $T\to T_c$ and we derive an explicit formula for the amplitude of $\ln |1-\frac{T}{T_c}$. From this result we demonstrate that if the $E_2\mathrm{}\left(j\right)\mathrm{}$ are considered to be independent random variables with a distribution function $P\left(E_2\right)$ which is not a $\delta$ function, then, for almost all lattices constructed from $P\left(E_2\right)$ the amplitude of the logarithmic singularity vanishes as $n\to \infty$.