In previous works, we showed that the internal DLA cluster on \Z^d with t particles is a.s. spherical up to a maximal error of O(\log t) if d=2 and O(\sqrt{ \log t}) if d > 2. This paper addresses "average error": in a certain sense, the average deviation of internal DLA from its mean shape is of constant order when d=2 and of order r^{ 1-d/2} (for a radius r cluster) in general. Appropriately normalized, the fluctuations (taken over time and space) scale to a variant of the Gaussian free field.
Categories: math.PR cond-mat.stat-mech math-ph math.AP math.MP