We describe a matrix analogy of the log-normal random walk, in the large N (size of the matrix) limit. In particular, we present an exact result for the infinite product of random matrices, corresponding to the multiplicative diffusion triggered by Ginibre--Girko ensemble. We observe the emergence of a ``topological phase transition'', when a hole develops in the complex eigenvalue spectrum, after some critical diffusion time \tau _crit is reached.

PACS numbers: 05.40.+j, 05.45+b, 05.70.Fh, 11.15.Pg