  We address a long-standing debate regarding the finite-size scaling of the
Ising model in high dimensions, by introducing a random-length random walk
model, which we then study rigorously. We prove that this model exhibits the
same universal FSS behaviour previously conjectured for the self-avoiding walk
and Ising model on finite boxes in high-dimensional lattices. Our results show
that the mean walk length of the random walk model controls the scaling
behaviour of the corresponding Green's function. We numerically demonstrate the
universality of our rigorous findings by extensive Monte Carlo simulations of
the Ising model and self-avoiding walk on five-dimensional hypercubic lattices
with free and periodic boundaries.
