  The Ising model exhibits qualitatively different properties in hyperbolic
space in comparison to its flat space counterpart. Due to the negative
curvature, a finite fraction of the total number of spins reside at the
boundary of a volume in hyperbolic space. As a result, boundary conditions play
an important role even when taking the thermodynamic limit. We investigate the
bulk thermodynamic properties of the Ising model in two and three dimensional
hyperbolic spaces using Monte Carlo and high and low-temperature series
expansion techniques. To extract the true bulk properties of the model in the
Monte Carlo computations, we consider the Ising model in hyperbolic space with
periodic boundary conditions. We compute the critical exponents and critical
temperatures for different tilings of the hyperbolic plane and show that the
results are of mean-field nature. We compare our results to the 'asymptotic'
limit of tilings of the hyperbolic plane: the Bethe lattice, explaining the
relationship between the critical properties of the Ising model on Bethe and
hyperbolic lattices. Finally, we analyze the Ising model on three dimensional
hyperbolic space using Monte Carlo and high-temperature series expansion. In
contrast to recent field theory calculations, which predict a non-mean-field
fixed point for the ferromagnetic-paramagnetic phase-transition of the Ising
model on three-dimensional hyperbolic space, our computations reveal a
mean-field behavior.
