  We describe a hierarchy of stochastic boundary conditions (SBCs) that can be
used to systematically eliminate finite size effects in Monte Carlo simulations
of Ising lattices. For an Ising model on a $100 \times 100$ square lattice, we
measured the specific heat, the magnetic susceptibility, and the spin-spin
correlation using SBCs of the two lowest orders, to show that they compare
favourably against periodic boundary conditions (PBC) simulations and
analytical results. To demonstrate how versatile the SBCs are, we then
simulated an Ising lattice with a magnetized boundary, and another with an open
boundary, measuring the magnetization, magnetic susceptibility, and
longitudinal and transverse spin-spin correlations as a function of distance
from the boundary.
