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The flux of particles in a one-dimensional Fleming-Viot process

Éric Brunet, Bernard Derrida

TL;DR

This work analyzes the Fleming-Viot process with absorbing origin, showing that in the large-$N$ limit the particle density becomes deterministic and akin to a quasi-stationary distribution, while providing an exact finite-time solution on the positive integers for arbitrary initial data. It reveals a Fisher-KPP–like one-parameter family of steady states, a Bramson-type logarithmic shift in long-time behavior, and a pulled–pushed transition induced by a near-origin modification of the diffusion, extended to the continuum half-line. A cut-off approximation is employed to predict leading large-$N$ corrections to the absorbed-flux, yielding $(\ln N)^{-2}$ scaling in the pulled regime and power-law scaling in the pushed regime, with a detailed crossover in the critical regime. The results connect explicit solvability with front-formation phenomena, offering precise predictions for finite-size effects and potential extensions to simulations and large-deviation analyses. The findings have broad relevance for understanding interacting particle systems with absorption and for exploring genealogies and fluctuations in quasi-stationary settings.

Abstract

The Fleming-Viot process describes a system of $N$ particles diffusing on a graph with an absorbing site. Whenever one of the particles is absorbed, it is replaced by a new particle at the position of one of the $N-1$ remaining particles. Here we consider the case where the particles lie on the semi-infinite line with a biased diffusion towards the origin which is the absorbing site. In the large $N$ limit, the evolution of the density becomes deterministic and has a number of characteristics similar to the Fisher-KPP equation: a one-parameter family of steady state solutions, dependence of the long time asymptotics on the initial conditions, Bramson logarithmic shift, etc. One noticeable difference, however, is that in the Fleming-Viot case, the solution can be computed explicitly for arbitrary initial conditions and at an arbitrary time. By modifying the diffusion rule near the origin, one can produce a transition in the flux of absorbed particles, very similar to the pushed-pulled transition in travelling waves. Lastly, using a cut-off approximation (which is known to be correct in the theory of travelling waves), we derive a number of predictions for the leading large $N$ correction of the flux of absorbed particles.

The flux of particles in a one-dimensional Fleming-Viot process

TL;DR

This work analyzes the Fleming-Viot process with absorbing origin, showing that in the large- limit the particle density becomes deterministic and akin to a quasi-stationary distribution, while providing an exact finite-time solution on the positive integers for arbitrary initial data. It reveals a Fisher-KPP–like one-parameter family of steady states, a Bramson-type logarithmic shift in long-time behavior, and a pulled–pushed transition induced by a near-origin modification of the diffusion, extended to the continuum half-line. A cut-off approximation is employed to predict leading large- corrections to the absorbed-flux, yielding scaling in the pulled regime and power-law scaling in the pushed regime, with a detailed crossover in the critical regime. The results connect explicit solvability with front-formation phenomena, offering precise predictions for finite-size effects and potential extensions to simulations and large-deviation analyses. The findings have broad relevance for understanding interacting particle systems with absorption and for exploring genealogies and fluctuations in quasi-stationary settings.

Abstract

The Fleming-Viot process describes a system of particles diffusing on a graph with an absorbing site. Whenever one of the particles is absorbed, it is replaced by a new particle at the position of one of the remaining particles. Here we consider the case where the particles lie on the semi-infinite line with a biased diffusion towards the origin which is the absorbing site. In the large limit, the evolution of the density becomes deterministic and has a number of characteristics similar to the Fisher-KPP equation: a one-parameter family of steady state solutions, dependence of the long time asymptotics on the initial conditions, Bramson logarithmic shift, etc. One noticeable difference, however, is that in the Fleming-Viot case, the solution can be computed explicitly for arbitrary initial conditions and at an arbitrary time. By modifying the diffusion rule near the origin, one can produce a transition in the flux of absorbed particles, very similar to the pushed-pulled transition in travelling waves. Lastly, using a cut-off approximation (which is known to be correct in the theory of travelling waves), we derive a number of predictions for the leading large correction of the flux of absorbed particles.
Paper Structure (18 sections, 113 equations, 1 figure)

This paper contains 18 sections, 113 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Fleming-Viot process on positive integers
  3. Steady state solutions
  4. The exact solution of for arbitrary initial conditions
  5. Long time asymptotics
  6. The transition between the pulled and pushed cases
  7. The Fleming-Viot process on the real line
  8. Steady states solutions
  9. Long time asymptotics for arbitrary initial conditions
  10. Pulled-pushed transition on the real line
  11. Cut-off prediction
  12. The pulled case
  13. The critical case
  14. The pushed case
  15. The critical regime
  16. ...and 3 more sections

Figures (1)

  • Figure 1: $\chi^2$ as a function of $c$. The correction to the absorption rate is then given by \ref{['121']}.