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  1. ; (, Communications in Mathematical Physics)
    This article shows that for a large class of discrete periodic Schrödinger operators, most wavefunctions resemble Bloch states. More precisely, we prove quantum ergodicity for a family of periodic Schrödinger operators H on periodic graphs. This means that most eigenfunctions of H on large finite periodic graphs are equidistributed in some sense, hence delocalized. Our results cover the adjacency matrix on Z^d, the triangular lattice, the honeycomb lattice, Cartesian products, and periodic Schrödinger operators on Z^d. The theorem applies more generally to any periodic Schrödinger operator satisfying an assumption on the Floquet eigenvalues. 
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  2. ; ; ; (, Communications in Mathematical Physics)
    Let G be a random d-regular graph on n vertices. We prove that for every constant a>0, with high probability every eigenvector of the adjacency matrix of G with eigenvalue sufficiently small has Omega(n/polylog(n)) nodal domains. 
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