Journal of Statistical Physics, Vol. 77, Nos. 3/4, 1994. p. 783-806.
The secular equations for finite tori of lattice sites are obtained by computer expansion of determinants for a hard-particle lattice gas. The secular equations yield all of the thermodynamic functions for finite systems and the beginning terms of activity expansions for all of the eigenvalues of the infinite system. The same secular equation that yields the low-density series for the equation of state of the infinite-circumference system also yields the beginning terms in the high-density expansion. As examples we treat two hard-particle lattice gases in two dimensions, the lattice gas with nearest-neighbor exclusion (which has a second-order transition), and the case of dimers (which is analytic all the way to close packing).