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Dyson’s nonintersecting Brownian motions with a few outliers. (English) Zbl 1166.60048

Consider \(n\) nonintersecting real Brownian bridges all starting from the origin at time \(t=0\) and returning to the origin at time \(t=1\) (Dyson Brownian motions). For large n, the average mean density of the particles has its support, for each \(0 < t < 1\) in the interval \((-c,c)\) where \(c=\sqrt{2nt(1-t)}\). The Airy process \(A(t)\) (introduced by M. Prähofer and H. Spohn [J. Stat. Phys. 108, No. 5–6, 1071–1106 (2002; Zbl 1025.82010)]) is defined as the motion of these nonintersecting Brownian bridges for large n, but viewed from an observer on the (right-hand) edge curve
\[ C: \{y=\sqrt{2nt(1-t)} > 0\} \]
with time and space appropriately rescaled. In this new scale the Airy process describes the fluctuations of the Brownian particles near the edge curve \(C\).
The authors study the case where among those n paths, \(0 \leq r \leq n\) are forced to reach a given final target \(a=\rho_0\sqrt{n/2}\), say, while the remaining \(n-r\) particles still return to the origin. The main result states that for large \(n\) no new process appears as long as we have not yet reached the time where the tangent to \(C\) passing through \(a\) touches \(C\). At the point of tangency the fluctuations obey a new statistics, which is called the Airy process with \(r\) outliers or \(r\)-Airy process. The \(r\)-Airy process is an extension of the Airy process, but it is in contrast to the Airy process not stationary. The logarithm of the probability that at time \(t\) the cloud does not exceed \(x\) is given by the Fredholm determinant of a new kernel (extending the Airy kernel), and it satisfies a nonlinear PDE in \(x\) and \(t\), from which the asymptotic behaviour of the process can be deduced for \(t\to-\infty\). This kernel is closely related to one found by Baik, Ben Arous, and Péché in the context of multivariate statistics.

MSC:

60J65 Brownian motion
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
82B10 Quantum equilibrium statistical mechanics (general)
60J60 Diffusion processes

Citations:

Zbl 1025.82010
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References:

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