  The Schelling model, introduced by Schelling in 1969 as a model for
residential segregation in cities, describes how populations of multiple types
self-organize to form homogeneous clusters of one type. In this model, vertices
in an $N$-dimensional lattice are initially assigned types randomly. As time
evolves, the type at a vertex $v$ has a tendency to be replaced with the most
common type within distance $w$ of $v$. We present the first mathematical
description of the dynamical scaling limit of this model as $w$ tends to
infinity and the lattice is correspondingly rescaled. We do this by deriving an
integro-differential equation for the limiting Schelling dynamics and proving
almost sure existence and uniqueness of the solutions when the initial
conditions are described by white noise. The evolving fields are in some sense
very "rough" but we are able to make rigorous sense of the evolution. In a key
lemma, we show that for certain Gaussian fields $h$, the supremum of the
occupation density of $h-\phi$ at zero (taken over all $1$-Lipschitz functions
$\phi$) is almost surely finite, thereby extending a result of Bass and Burdzy.
In the one dimensional case, we also describe the scaling limit of the limiting
clusters obtained at time infinity, thereby resolving a conjecture of Brandt,
Immorlica, Kamath, and Kleinberg.
