Abstract
Two classes of interacting particle systems on mathbb {Z} are shown to be Pfaffian point processes, at any fixed time and for all deterministic initial conditions. The first comprises coalescing and branching random walks, the second annihilating random walks with pairwise immigration. Various limiting Pfaffian point processes on mathbb {R} are found by diffusive rescaling, including the point set process for the Brownian web and Brownian net.
Highlights
Introduction and Statement of Key ResultsInteracting particle systems of reaction–diffusion type have been an object of mathematical investigation for a long time, see e.g. [9]
This paper continues the study in [6], where it was shown that systems of instantaneously annihilating or coalescing random walks on Z are Pfaffian point processes, at any fixed time and for all deterministic initial conditions
The purpose in this paper is to describe two additional mechanisms that preserve this Pfaffian property
Summary
Interacting particle systems of reaction–diffusion type have been an object of mathematical investigation for a long time, see e.g. [9]. This paper continues the study in [6], where it was shown that systems of instantaneously annihilating or coalescing random walks on Z are Pfaffian point processes, at any fixed time and for all deterministic initial conditions. For any initial condition η ∈ {0, 1}Z for the ARPI model, and at any fixed time t ≥ 0, the variable ηt is a Pfaffian point process with kernel K given, for y < z, by. 4, which we call the Brownian firework, pairs of particles are immigrated only at the origin at an infinite rate This has a steady-state X∞ (c) where the immigration and the annihilation balance each other: X∞ (c) is a Pfaffian point process on R\{0} with kernel K(∞c) of the form.
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