  We present a modified Brownian motion model for random matrices where the
eigenvalues (or levels) of a random matrix evolve in "time" in such a way that
they never cross each other's path. Also, owing to the exact integrability of
the level dynamics, we incorporate long-time recurrences into the random walk
problem underlying the Brownian motion. From this model, we derive the Coulomb
interaction between the two eigenvalues. We further show that the Coulomb gas
analogy fails if the confining potential, $V(E)$ is a transcendental function
such that there exist orthogonal polynomials with weighting function, $\exp
[-\beta E]$, where $\beta $ is a symmetry parameter.
