  Recent experiments have shown that the superconducting energy gap in some
cuprates is spatially inhomogeneous. Motivated by these experiments, and using
exact diagonalization of a model d-wave Hamiltonian, combined with Monte Carlo
simulations of a Ginzburg-Landau free energy functional, we have calculated the
single-particle density of states LDOS$(\omega,r)$ of a model high-T$_c$
superconductor as a function of temperature. Our calculations include both
quenched disorder in the pairing potential and thermal fluctuations in both
phase and amplitude of the superconducting gap. Most of our calculations assume
two types of superconducting regions: $\alpha$, with a small gap and large
superfluid density, and $\beta$, with the opposite. If the $\beta$ regions are
randomly embedded in an $\alpha$ host, the LDOS on the $\alpha$ sites still has
a sharp coherence peak at $T = 0$, but the $\beta$ component does not, in
agreement with experiment. An ordered arrangement of $\beta$ regions leads to
oscillations in the LDOS as a function of energy. The model leads to a
superconducting transition temperature $T_c$ well below the pseudogap
temperature $T_{c0}$, and has a spatially varying gap at very low $T$, both
consistent with experiments in underdoped Bi2212. Our calculated
LDOS$(\omega,r)$ shows coherence peaks for $T < T_c$, which disappear for $T >
T_c$, in agreement with previous work considering phase but not amplitude
fluctuations in a homogeneous superconductor. Well above $T_c$, the gap in the
LDOS disappears.
