We consider the relation between nonequilibrium growth processes associated with viscous fingering (e.g. diffusion limited aggregation, (DLA) and Saffman-Taylor dynamics in Hele-Shaw cells) and the eigenvalue distribution of normal matrices model (Complex matrices, M and M' which commute). Viscous fingering is a classical paradigm model for pattern formation, and a long standing problem is to understand its fractal properties. On the other hand, the normal matrix model describes, apparently, very remote fields, for instance, Quantum Hall Effect in nonuniform magnetic field [1], The Toda lattice hierarchy, and two-dimensional quantum gravity. Yet, it is possible to show that in the classical (dispersionless) limit (h→0), the normal matrix model describes precisely viscous fingering with zero surface tension. Using this relation we construct an algebro-geometric, genus one, solution for the two dimensional Toda lattice equation [2]. This solution, analogous to Gurevich-Pitaevskii solution of the KdV equation, enables us to give a detailed account of the tip-splitting process. Tip splitting is the basic ingredient characterizing viscous, and its properties allows one to deduce, e.g., the fractal dimension of DLA and Saffman-Taylor droplets [3].

[1] O. Agam, E. Bettelheim, P. Wiegmann, and A. Zabrodin, Phys. Rev. Lett. 88, 236801 (2002).
[2] E. Bettelheim and O. Agam, in preparation.
[3] T.C. Halsey , Phys. Rev. Let. 72, 1228 (1994)