The Bakry-Emery gradient estimate for the labeled Dyson space holds exactly when the inverse temperature beta is at least 1 and fails for every beta below 1, a sharp phase transition between curvature and gradient estimates.
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A phase transition in the Bakry-\'Emery gradient estimate for Dyson Brownian motion
The Bakry-Emery gradient estimate for the labeled Dyson space holds exactly when the inverse temperature beta is at least 1 and fails for every beta below 1, a sharp phase transition between curvature and gradient estimates.