Configurational transition in a Fleming-Viot-type model and probabilistic interpretation of Laplacian eigenfunctions

TL;DR

This paper investigates a multi-type Brownian particle system with death and branching in a confined rectangle, revealing a transition in stationary distribution linked to Laplacian eigenfunctions and the system's geometry.

Contribution

It introduces a probabilistic model connecting particle configurations with Laplacian eigenfunctions and identifies a geometric transition related to eigenvalues.

Findings

Stationary distribution corresponds to the m-th Laplacian eigenfunction in elongated rectangles.

A configurational transition occurs when the aspect ratio crosses a critical value.

The transition point relates to the m-th eigenvalue of the Laplacian with rectangular boundaries.

Abstract

We analyze 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 the two dimensional box, whose boundaries act as the sink of Brownian particles. The branching rate matches the death rate so that the total number of particles is kept constant. In the case of m types of particles in the rectangular box of size a,b and elongated shape $a\gg b$ we observe that the stationary distribution of particles corresponds to the m-th 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 m-th eigenvalue of the Laplacian with rectangular boundaries.