Dynamics of Phase Boundary with Particle Annihilation

Infinitely many particles of two types ("plus" and "minus") jump randomly along the one-dimensional lattice $\mathbf{Z}_{\varepsilon}=\varepsilon\mathbf{Z}$. Annihillations occur when two particles of different time occupy the same site. Assuming that at time $t=0$ all "minus" particles are placed on the left of the origin and all "plus" particles are on the right of it, we study evolution of $\beta_\varepsilon(t)$, the boundary between two types. We prove that in large density limit $\epsilon\to 0$ the boundary $\beta_\varepsilon(t)$ converges to a deterministic limit. This particle system can be interpreted as a microscopic model of price formation on economic markets with large number of players.