  The four dimensional Gaussian random field Ising magnet is investigated
numerically at zero temperature, using samples up to size $64^4$, to test
scaling theories and to investigate the nature of domain walls and the
thermodynamic limit. As the magnetization exponent $\beta$ is more easily
distinguishable from zero in four dimensions than in three dimensions, these
results provide a useful test of conventional scaling theories. Results are
presented for the critical behavior of the heat capacity, magnetization, and
stiffness. The fractal dimensions of the domain walls at criticality are
estimated. A notable difference from three dimensions is the structure of the
spin domains: frozen spins of both signs percolate at a disorder magnitude less
than the value at the ferromagnetic to paramagnetic transition. Hence, in the
vicinity of the transition, there are two percolating clusters of opposite
spins that are fixed under any boundary conditions. This structure changes the
interpretation of the domain walls for the four dimensional case. The scaling
of the effect of boundary conditions on the interior spin configuration is
found to be consistent with the domain wall dimension. There is no evidence of
a glassy phase: there appears to be a single transition from two ferromagnetic
states to a single paramagnetic state, as in three dimensions. The slowing down
of the ground state algorithm is also used to study this model and the links
between combinatorial optimization and critical behavior.
