We calculate the large-
q series expansion for the ground-state degeneracy (= exponent of the ground-state entropy) per site of the
q-state Potts antiferromagnet on the $(3\ensuremath{\cdot}{12}^{2})$ lattice, to order
O(y19), where $y=1/(q\ensuremath{-}1)$. We note a remarkable agreement, to
O(y18), between this series and a rigorous lower bound derived recently.
We study the ground state degeneracy per site (exponent of the ground state entropy) $W(\ensuremath{\Lambda}{,(L}_{x}=\ensuremath{\infty})\ifmmode\times\else\texttimes\fi{}{L}_{y},q)$ for the
q-state Potts antiferromagnet on infinitely long strips with width
Ly of 2D lattices \ensuremath{\Lambda} with free and periodic boundary conditions in the
y direction, denoted
FBCy and
PBCy. We show that the approach of
W to its 2D thermodynamic limit as
Ly increases is quite rapid; for moderate values of
q and ${L}_{y}\ensuremath{\simeq}4,$ $W(\ensuremath{\Lambda}{,(L}_{x}=\ensuremath{\infty})\ifmmode\times\else\texttimes\fi{}{L}_{y},q)$ is within about 5% and ${O(10}^{\ensuremath{-}3})$ of the 2D value $W(\ensuremath{\Lambda}{,(L}_{x}=\ensuremath{\infty})\ifmmode\times\else\texttimes\fi{}{(L}_{y}=\ensuremath{\infty}),q)$ for
FBCy and
PBCy, respectively. The approach of
W to the 2D thermodynamic limit is proved to be monotonic (nonmonotonic) for
FBCy (PBCy). It is noted that ground state entropy determinations on infinite strips can be used to obtain the central charge for cases with critical ground states.
We present the result of calculations of the Witten index for a supersymmetric lattice model on lattices of various type and size. Because the model remains supersymmetric at finite lattice size, the Witten index can be calculated using row-to-row transfer matrices and the calculations are similar to calculations of the partition function at negative activity −1. The Witten index provides a lower bound on the number of ground states. We find strong numerical evidence that the Witten index grows exponentially with the number of sites of the lattice, implying that the model has extensive entropy in the ground state.