A reflected Brownian motion in the Gelfand-Tsetlin cone is used to construct Dyson's process of non-colliding Brownian motions. The key step of the construction is to consider two interlaced families of Brownian paths with paths belonging to the second family reflected off paths belonging to the first. Such families of paths are known to arise in the Arratia flow of coalescing Brownian motions. A determinantal formula for the distribution of coalescing Brownian motions is presented.
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