## Electronic Journal of Probability

### Nonintersecting paths with a staircase initial condition

#### Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465062382

Digital Object Identifier
doi:10.1214/EJP.v17-1902

Mathematical Reviews number (MathSciNet)
MR2959066

Zentralblatt MATH identifier
1266.60089

Subjects
Primary: 60G55: Point processes

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

#### Citation

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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