In this paper, we provide the exact expression for the coefficients in the low-temperature series expansion of the partition function of the two-dimensional Ising model on the infinite square lattice. This is equivalent to exact determination of the number of spin configurations at a given energy. With these coefficients, we show that the ferromagnetic--to--paramagnetic phase transition in the square lattice Ising model can be explained through equivalence between the model and the perfect gas of energy clusters model, in which the passage through the critical point is related to the complete change in the thermodynamic preferences on the size of clusters. The combinatorial approach reported in this article is very general and can be easily applied to other lattice models.
We analyze the ensemble of surfaces surrounding critical clusters at T=
Tc in the 3D Ising model. We find that
Ng(A), the number of surfaces of genus g and area A, behaves as
Ax(g)${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\mu}}\mathit{A}}$. We show that \ensuremath{\mu} is constant and x(g) is approximately linear; the sum
tsumg Ng(A) scales as a power of A. The cluster volume is proportional to its surface area. We discuss similar reuslts for the ordinary spin clusters of the 3D Ising model and for 3D bond percolation.