Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ben-Avraham, D., Brunet, E.: On the relation between one-species diffusion-limited coalescence and annihilation in one dimension. J. Phys. A 38(15), 3247–3252 (2005)
Billingsley, P.: Convergence of Probability Measures. Wiley, Hoboken (1968)
Borodin, A., Corwin, I., Sasamoto, T.: From duality to determinants for q-TASEP and ASEP. Ann. Probab. 42(6), 2314–2382 (2014)
Borodin, A., Sinclair, C.D.: The Ginibre ensemble of real random matrices and its scaling limits. Commun. Math. Phys. 291(1), 177–224 (2009)
Borodin, A., Poplavskyi, M., Sinclair, C.D., Tribe, R., Zaboronski, O.: Erratum to: the Ginibre ensemble of real random matrices and its scaling limits. Commun. Math. Phys. 346, 1051 (2016)
Connaughton, C., Rajesh, R., Tribe, R., Zaboronski, O.: Non-equilibrium phase diagram for a model with coalescence, evaporation and deposition. J. Stat. Phys. 152(6), 1115–1144 (2013)
Derrida, B., Bray, A.J., Godreche, C.: Nontrivial exponents in the zero temperature dynamics of the 1D Ising and Potts models. J. Phys. A 27(11), L357–L361 (1994)
Derrida, B., Zeitak, R.: Distribution of domain sizes in the zero temperature Glauber dynamics of the 1d Potts model. Phys. Rev. E 54, 2513–2525 (1996)
Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, p. x+534. Wiley, New York (1986). ISBN: 0-471-08186-8
Felderhof, B.U.: Spin relaxation of the Ising chain, Reports on Mathematical Physics 1, 215 (1970), and Note on spin relaxation of the Ising chain. Reports on Mathematical Physics 2, 151–152 (1971)
Forrester, P.J., Nagao, T.: Eigenvalue statistics of the real Ginibre ensemble. Phys. Rev. Lett. 99(5), 050603 (2007)
Garrod, B., Tribe, R., Zaboronski, O.: Examples of interacting particle systems on \(\mathbb{Z}\) as Pfaffian point processes II—coalescing branching random walks and annihilating random walks with immigration
Garrod, B.: Warwick Thesis (2016)
Glauber, R.J.: Time-dependent statistics of the Ising model. J. Math. Phys. 4, 294–307 (1963)
Henkel, M.: Classical and Quantum Nonlinear Integrable Systems: Theory and Applications, pp. 256–287. Institute of Physics Publishing Ltd, Bristol and Philadelphia (2003)
Le Doussal, P., Monthus, C.: Reaction diffusion models in one dimension with disorder. Phys. Rev. E (3) 60(2, part A), 1212–1238 (1999)
Poplavskyi, M., Tribe, R., Zaboronski, O.: On the distribution of the largest real eigenvalue for the real Ginibre ensemble. Ann. Appl. Probab. 27(3), 1395–1413 (2017)
Rider, B., Sinclair, C.D.: Extremal laws for the real Ginibre ensemble. Ann. Appl. Probab. 24(4), 1621–1651 (2014)
Sommers, H.-J., Wieczorek, W.: General eigenvalue correlations for the real Ginibre ensemble. J. Phys. A 41(40), 405003 (2008)
Soshnikov, A.: Determinantal random fields. In: Francoise, J.-P., Naber, G., Tsun, T.S. (eds.) Encyclopedia of Mathematical Physics, vol. 2, pp. 47–53. Elsevier, Oxford (2006)
Stembridge, J.R.: Non-intersecting paths, Pfaffians and plane partitions. Adv. Math. 83, 96–131 (1990)
Tracy, C.A., Widom, H.: On orthogonal and symplectic matrix ensembles. Commun. Math. Phys. 177, 727–754 (1996)
Tribe, R., Zaboronski, O.: Pfaffian formulae for one dimensional coalescing and annihilating systems. Electron. J. Probab. 16(76), 2080–2103 (2011). MR2851057
Tribe, R., Yip, J., Zaboronski, O.: One dimensional annihilating and coalescing particle systems as extended Pfaffian point processes. Electron. Commun. Probab. 17(40), p. 7. (2012), and Erratum Electron. Commun. Probab. 20, paper no. 46, p. 2 (2015)
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Vadim Gorin.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), 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.
About this article
Cite this article
Garrod, B., Poplavskyi, M., Tribe, R.P. et al. Examples of Interacting Particle Systems on \(\mathbb {Z}\) as Pfaffian Point Processes: Annihilating and Coalescing Random Walks. Ann. Henri Poincaré 19, 3635–3662 (2018). https://doi.org/10.1007/s00023-018-0719-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-018-0719-x