Harmonic measure in the presence of a spectral gap

We study harmonic measure in finite graphs with an emphasis on expanders, that is, positive spectral gap. It is shown that if the spectral gap is positive then for all sets that are not too large the harmonic measure from a uniform starting point is not more than a constant factor of the uniform measure on the set. For large sets there is a tight logarithmic correction factor. We also show that positive spectral gap does not allow for a fixed proportion of the harmonic measure of sets to be supported on small subsets, in contrast to the situation in Euclidean space. The results are quantitative as a function of the spectral gap, and apply also when the spectral gap decays to 0 as the size of the graph grows to infinity. As an application we consider a model of diffusion limited aggregation, or DLA, on finite graphs, obtaining upper bounds on the growth rate of the aggregate.