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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2601.16029 |
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| _version_ | 1866912841316958208 |
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| author | Brunet, Éric Derrida, Bernard |
| author_facet | Brunet, Éric Derrida, Bernard |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_16029 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The flux of particles in a one-dimensional Fleming-Viot process Brunet, Éric Derrida, Bernard Statistical Mechanics 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. |
| title | The flux of particles in a one-dimensional Fleming-Viot process |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2601.16029 |