The Annals of Applied Probability

Scaling limit for Brownian motions with one-sided collisions

Patrik L. Ferrari, Herbert Spohn, and Thomas Weiss

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We consider Brownian motions with one-sided collisions, meaning that each particle is reflected at its right neighbour. For a finite number of particles a Schütz-type formula is derived for the transition probability. We investigate an infinite system with periodic initial configuration, that is, particles are located at the integer lattice at time zero. The joint distribution of the positions of a finite subset of particles is expressed as a Fredholm determinant with a kernel defining a signed determinantal point process. In the appropriate large time scaling limit, the fluctuations in the particle positions are described by the $\mathrm{Airy}_{1}$ process.

Article information

Ann. Appl. Probab., Volume 25, Number 3 (2015), 1349-1382.

First available in Project Euclid: 23 March 2015

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J65: Brownian motion [See also 58J65]

Brownian motion one-sided collision Airy$_{1}$ process Fredholm determinant periodic initial configuration


Ferrari, Patrik L.; Spohn, Herbert; Weiss, Thomas. Scaling limit for Brownian motions with one-sided collisions. Ann. Appl. Probab. 25 (2015), no. 3, 1349--1382. doi:10.1214/14-AAP1025.

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