  A statistical theory of two-dimensional Laplacian growths is formulated from
first-principles. First the area enclosed by the growing surface is mapped
conformally to the interior of the unit circle, generating a set of dynamically
evolving quasi-particles. Then it is shown that the evolution of a
surface-tension-free growing surface is Hamiltonian. The Hamiltonian
formulation allows a natural extension of the formalism to growths with either
isotropic or anisotropic surface tension. It is shown that the curvature term
can be included as a surface energy in the Hamiltonian that gives rise to
repulsion between the quasi-particles and the surface. This repulsion prevents
cusp singularities from forming along the surface at any finite time and
regularizes the growth. An explicit example is computed to demonstrate the
regularizing effect. Noise is then introduced as in traditional statistical
mechanical formalism and a measure is defined that allows analysis of the
spatial distribution of the quasi-particles. Finally, a relation is derived
between this distribution and the growth probability along the growing surface.
Since the spatial distribution of quasi-particles flows to a stable limiting
form, this immediately translates into predictability of the asymptotic
morphology of the surface. An exactly-solvable class of arbitrary initial
conditions is analysed explicitly.
