Electronic Journal of Probability

Nonintersecting paths with a staircase initial condition

Jonathan Breuer and Maurice Duits

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We consider an ensemble of $N$ discrete nonintersecting paths starting from equidistant points and ending at consecutive integers. Our first result is an explicit formula for the correlation kernel that allows us to analyze the process as $N\to \infty$.   In that limit we obtain a new general class of kernels describing the local correlations close to the equidistant starting points. As the distance between the starting points goes to infinity, the correlation kernel converges to that of a single random walker. As the distance to the starting line increases, however, the local correlations converge to the sine kernel. Thus, this class interpolates between the sine kernel and an ensemble of independent particles. We also compute the scaled simultaneous limit, with both the distance between particles and the distance to the starting line going to infinity, and obtain a process with number variance saturation, previously studied by Johansson.

Article information

Electron. J. Probab., Volume 17 (2012), paper no. 60, 24 pp.

Accepted: 3 August 2012
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G55: Point processes

Random non-intersecting paths Determinantal point processes random tilings

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Breuer, Jonathan; Duits, Maurice. Nonintersecting paths with a staircase initial condition. Electron. J. Probab. 17 (2012), paper no. 60, 24 pp. doi:10.1214/EJP.v17-1902. https://projecteuclid.org/euclid.ejp/1465062382

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