Pfaffian formulae for one dimensional coalescing and annihilating systems

Abstract

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.

26 pages, no figures. Final version: connection between one-dimensional distributions of annihilating Brownian motions and real eigenvalues in real Ginibre ensemble spotted by the referee is stated explicitly. Conjecture relating the eigenvalue processes to annihilating Brownian motions is stated. References added