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David Aldous studies probability theory and random combinatorial structures. His random-walk construction of a uniform spanning tree keeps the first-entrance edge for each newly visited vertex of a finite connected graph.^[ref]

Those edges necessarily form a tree, but their uniform distribution requires a separate argument. The construction permits properties of random trees—degrees, leaves and diameter—to be studied through random walks. Its running time is tied to the walk's cover time, giving a direct link between a stochastic exploration process and a combinatorial sampling algorithm.^[ref]