Brownian Gaussian

Suppose a particle on an integer lattice can move one step in any of the cardinal directions with equal probabilities on each time step. The asymptotic distribution of the particle's location approaches a circular Gaussian. Two mnemonics suggest this: The directions are a kernel for a Laplacian, and repeated application creates a Gaussian blur. Separating on each dimension, the location distribution is binomial, which in turn asymptotically approaches a Gaussian.

How does one measure the deviation of the current iteration of the Laplacian from the ideal Gaussian. It is likely that some moment of the distribution captures the cross-shapedness which asymptotically approaches circularity.