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We discuss the connection between the expansion of small sets in graphs, andthe Schatten norms of their adjacency matrix. In conjunction with a variant ofthe Azuma inequality for uniformly smooth normed spaces, we deduce improvedbounds on the small set isoperimetry of Abelian Alon-Roichman random Cayleygraphs.

On the Banach space valued Azuma inequality and small set isoperimetry of Alon-Roichman graphs