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In previous works, we showed that the internal diffusion-limited aggregation (DLA) cluster on with particles is almost surely spherical up to a maximal error of if and if . This paper addresses average error: in a certain sense, the average deviation of internal DLA from its mean shape is of constant order when and of order (for a radius cluster) in general. Appropriately normalized, the fluctuations (taken over time and space) scale to a variant of the Gaussian free field.
Kaimanovich and Masur showed that a random walk on the mapping class group for an initial distribution whose support generates a nonelementary subgroup when projected into Teichmüller space converges almost surely to a point in the space of projective measured foliations on the surface. This defines a harmonic measure on . Here, we show that when the initial distribution has finite support, the corresponding harmonic measure is singular with respect to the natural Lebesgue measure class on .
We consider the nonlinear Schrödinger equation in dimension and in the mass supercritical and energy subcritical range . For initial data with radial symmetry, we prove a universal upper bound on the blow-up speed. We then prove that this bound is sharp and attained on a family of collapsing ring blow-up solutions first formally predicted in Fibich et al.