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Pfaffian formulae for one dimensional coalescing and annihilating systems. (English) Zbl 1244.60097
Summary: The paper considers instantly coalescing, or instantly annihilating, systems of one-dimensional Brownian particles on the real line. Under maximal entrance laws, the distribution of the particles at a fixed time is shown to be Pfaffian point processes closely related to the Pfaffian point process describing one dimensional distribution of real eigenvalues in the real Ginibre ensemble of random matrices. As an application, an exact large time asymptotic for the $$n$$-point density function for coalescing particles is derived.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60B20 Random matrices (probabilistic aspects) 82C22 Interacting particle systems in time-dependent statistical mechanics
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