CONFIGURATIONAL TRANSITION IN A FLEMING-VIOT-TYPE MODEL AND PROBABILISTIC-INTERPRETATION OF LAPLACIAN EIGENFUNCTIONS

We analyse and simulate a two-dimensional Brownian multi-type particle system with death and branching (birth) depending on the position of particles of different types. The system is confined in a two-dimensio nal box, whose boundaries act as the sink of Brownian particles. The b ranching rate matches the death rate so that the total number of parti cles is kept constant. In the case of m types of particle in a rectang ular box of size a x b and elongated shape a much greater than b we ob serve that the stationary distribution of particles corresponds to the mth Laplacian eigenfunction. For smaller elongations a > b we find a configurational transition to a new limiting distribution. The ratio a /b for which the transition occurs is related to the value of the mth eigenvalue of the Laplacian with rectangular boundaries.