Critical behavior of non-intersecting Brownian motions. (English) Zbl 1446.82062
Summary: We study \(n\) non-intersecting Brownian motions corresponding to initial configurations which have a vanishing density in the large \(n\) limit at an interior point of the support. It is understood that the point of vanishing can propagate up to a critical time, and we investigate the nature of the microscopic space-time correlations near the critical point and critical time. We show that they are described either by the Pearcey process or by the Airy line ensemble, depending on whether a simple integral related to the initial configuration vanishes or not. Since the Airy line ensemble typically arises near edge points of the macroscopic density, its appearance in the interior of the spectrum is surprising. We explain this phenomenon by showing that, even though there is no gap of macroscopic size near the critical point, there is with high probability a gap of mesoscopic size. Moreover, we identify a path which follows the \(\text{Airy}_2\) process.
MSC:
| 82C27 | Dynamic critical phenomena in statistical mechanics |
| 82C31 | Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics |
| 60J65 | Brownian motion |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 42B10 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
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