## Abstract

The mathematical properties of the exact solution of the hard-hexagon lattice gas model are investigated by using the Klein-Fricke theory of modular functions. In particular, it is shown that the order-parameter R and the reciprocal activity z$^{\prime}$ for the model can be expressed in terms of hauptmoduls that are associated with certain congruence subgroups of the full modular group $\Gamma $. Known modular equations are then used to prove that R(z$^{\prime}$) is an algebraic function of z$^{\prime}$. A connection is established between the singular points of this function and the geometrical properties of the icosahedron. Various algebraic and hypergeometric closed-form expressions are also derived for the order-parameter R(z$^{\prime}$) in terms of the fundamental modular invariant J. Next the simple relation $\Xi _{+}^{6}$ = (z$^{\prime}$)$^{-2}$[1 - 11z$^{\prime}$ - (z$^{\prime}$)$^{2}$]$^{-1}$R$^{9}$, (0 < z$^{\prime}$ < z$_{\text{c}}^{\prime}$) is derived, where $\Xi _{+}$ is the grand partition function per site and z$_{\text{c}}^{\prime}$ = ${\textstyle\frac{1}{2}}$(5$\surd $5 - 11) denotes the critical value of z$^{\prime}$. This important result forms the basis for a detailed analysis of the properties of the mean density $\rho $(z$^{\prime}$) and the isothermal compressibility $\kappa _{\text{T}}$(z$^{\prime}$) in the ordered region. It is shown that the mean density $\rho $(z$^{\prime}$) is a solution of the algebraic equation 3[1 - 11z$^{\prime}$ - (z$^{\prime}$)$^{2}$]$\rho ^{4}$ - [1 - 66z$^{\prime}$ - 11(z$^{\prime}$)$^{2}$]$\rho ^{3}$ - 15z$^{\prime}$(3 + z$^{\prime}$)$\rho ^{2}$ + 3z$^{\prime}$(4 + 3z$^{\prime}$)$\rho $ - z$^{\prime}$(1 + 2z$^{\prime}$) = 0. From this relation we obtain the following simple closed-form expressions for the inverse function z$^{\prime}$($\rho $) and the isothermal compressibility $\kappa _{\text{T}}$($\rho $): z$^{\prime}$($\rho $) = -$\frac{1}{2}$(2 - 3$\rho $)$^{-1}$(1 - $\rho $)$^{-3}$[(1 - 12$\rho $ + 45$\rho ^{2}$ - 66$\rho ^{3}$ + 33$\rho ^{4}$) + (-1 + 5$\rho $ - 5$\rho ^{2}$)$^{\frac{3}{2}}$(-1 + 9$\rho $ - 9$\rho ^{2}$)$^{\frac{1}{2}}$], and k$_{\text{B}}$ T$\rho \kappa _{\text{T}}$($\rho $) = $\frac{1}{15\rho}$[(1 - 2$\rho $)(-1 + 9$\rho $ - 9$\rho ^{2}$)$^{\frac{1}{2}}$(-1 + 5$\rho $ - 5$\rho ^{2}$)$^{-\frac{1}{2}}$ - (-1 + 9$\rho $ - 9$\rho ^{2}$)], where $\rho _{\text{c}}$ < $\rho \leq {\textstyle\frac{1}{3}}$, and $\rho _{\text{c}}$ = ${\textstyle\frac{1}{10}}$(5 - $\surd $5). The formula for $\kappa _{\text{T}}$($\rho $) is used to write the equation of state of the lattice gas pa$_{0}$/k$_{B}$ T = ln $\Xi _{+}$($\rho $) $\equiv \Gamma _{+}(\rho)$ in terms of a certain pseudo-elliptic integral, which is evaluated exactly to give a further closed-form expression for $\Gamma _{+}$($\rho $). (The quantity a$_{0}$ is the area of a unit cell in the lattice.) Finally, the properties of the hard-hexagon model in the disordered region 0 < $\rho $ < $\rho _{\text{c}}$ are studied with further modular equations derived by Hermite, and Klein and Fricke. This work leads to a closed-form expression for the generating function of the irreducible cluster sums $\beta _{l}$, l = 1, 2, 3, $\ldots $, and an asymptotic formula for the viral coefficients $\beta _{l}$ as l $\rightarrow \infty $. It is also proved that the radius of convergence $\rho _{\text{r}}$ of the virial series for the pressure p is given by $\rho _{\text{r}}$ = ${\textstyle\frac{1}{10}}\surd $5[(4$\surd $10 - 5$\surd $5 + 5) - $\surd $10(4$\surd $10 - 5$\surd $5 - 4$\surd $2 + 7)$^{\frac{1}{2}}$]$^{\frac{1}{2}}$. A striking feature of this result is that $\rho _{\text{r}}$ is less than the critical density $\rho _{\text{c}}$ = ${\textstyle\frac{1}{10}}$(5 - $\surd $5).