Annales Henri Poincaré

, Volume 19, Issue 12, pp 3635–3662 | Cite as

Examples of Interacting Particle Systems on \(\mathbb {Z}\) as Pfaffian Point Processes: Annihilating and Coalescing Random Walks

  • Barnaby Garrod
  • Mihail Poplavskyi
  • Roger P. Tribe
  • Oleg V. ZaboronskiEmail author
Open Access


A class of interacting particle systems on \(\mathbb {Z}\), involving instantaneously annihilating or coalescing nearest neighbour random walks, are shown to be Pfaffian point processes for all deterministic initial conditions. As diffusion limits, explicit Pfaffian kernels are derived for a variety of coalescing and annihilating Brownian systems. For Brownian motions on \(\mathbb {R}\), depending on the initial conditions, the corresponding kernels are closely related to the bulk and edge scaling limits of the Pfaffian point process for real eigenvalues for the real Ginibre ensemble of random matrices. For Brownian motions on \(\mathbb {R}_{+}\) with absorbing or reflected boundary conditions at zero, new interesting Pfaffian kernels appear. We illustrate the utility of the Pfaffian structure by determining the extreme statistics of the rightmost particle for the purely annihilating Brownian motions, and also computing the probability of overcrowded regions for all models.



B.G. supported by EPSRC Grant EP/H023364/1; M. P. and R. T. supported by EPSRC Grant No. RMAA3188; O.Z. supported by Leverhulme Trust Research Fellowship.


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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Barnaby Garrod
    • 1
  • Mihail Poplavskyi
    • 1
  • Roger P. Tribe
    • 1
  • Oleg V. Zaboronski
    • 1
    Email author
  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK

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