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Edge scaling limit of Dyson Brownian motion at equilibrium for general $\beta \geq 1$
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abstract
For general $\beta \geq 1$, we consider Dyson Brownian motion at equilibrium and prove convergence of the extremal particles to an ensemble of continuous sample paths in the limit $N \to \infty$. For each fixed time, this ensemble is distributed as the Airy$_\beta$ random point field. We prove that the increments of the limiting process are locally Brownian. When $\beta >1$ we prove that after subtracting a Brownian motion, the sample paths are almost surely locally $r$-H{\"o}lder for any $r<1-(1+\beta)^{-1}$. Furthermore for all $\beta \geq 1$ we show that the limiting process solves an SDE in a weak sense. When $\beta=2$ this limiting process is the Airy line ensemble.
Forward citations
Cited by 3 Pith papers
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Applications of optimal transport to Dyson Brownian Motions and beyond
A new optimal transport argument shows that Dyson Brownian motions with beta at least 2, and their scaling limits, have Brownian-like uniform modulus of continuity bounds with constants independent of the particle layer.
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Airy$_\beta$ line ensemble and its Laplace transform
The Airy_beta line ensemble is constructed for all beta>0 via explicit multi-time Laplace transform formulas, and it is shown to be the edge scaling limit of both the Dyson Brownian Motion and the Gaussian beta corner...
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A convergence framework for Airy$_\beta$ line ensemble via pole evolution
A pole-evolution characterization of the Airy_beta line ensemble yields convergence of DBM with general potentials, Laguerre, and Jacobi processes to the universal edge limit.
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