Bolthausen–Sznitman ancestry for a fixed-size selected population
For an integer $N$, the $N$-branching Brownian motion with selection ($N$-BBM) consists of $N$ particles on $\mathbb{R}$ that move as independent standard Brownian motions and each split into two at rate $1$, at the parent's position; immediately after each split a leftmost particle is removed, so that exactly $N$ particles remain, and the parent of every child is recorded. Let $X_1(t), \dots, X_N(t)$ be the positions and $Y_i(t) = X_i(t) - \min_j X_j(t)$. Berestycki and Tough proved that the offset process $(Y_1(t), \dots, Y_N(t))$ has a unique stationary distribution $\psi_N$. Brunet, Derrida, Mueller and Munier predicted that in models of this kind the genealogy of the population, on a time scale of order $(\log N)^3$, converges to the Bolthausen–Sznitman coalescent. Berestycki, Berestycki and Schweinsberg proved this for branching Brownian motion with negative drift absorbed at zero, in a near-critical regime in which the population stays of order $N$. For the $N$-BBM, Maillard proved that after time is rescaled by $(\log N)^3$, the suitably recentred position of the $\lceil \alpha N \rceil$-th rightmost particle, $0 < \alpha < 1$, converges in law to an explicit spectrally positive Lévy process.
Problem. Is there a constant $c > 0$ such that, with $a_N = c (\log N)^3$, the following holds for every integer $k \ge 2$ and every $T > 0$? Start the offset configuration with law $\psi_N$, run the $N$-BBM up to time $a_N T$, and choose $k$ distinct particles alive at that time uniformly without replacement. For $0 \le s \le T$ let $\Pi_{N,k,T}(s)$ be the partition of $\{1, \dots, k\}$ in which two sampled particles share a block when they have the same ancestor at time $a_N (T - s)$. Then, as $N \to \infty$, $$\bigl(\Pi_{N,k,T}(s)\bigr)_{0 \le s \le T} \Longrightarrow \bigl(\Pi_k(s)\bigr)_{0 \le s \le T}$$ in finite-dimensional distributions, where $\Pi_k$ is the standard Bolthausen–Sznitman coalescent on $\{1, \dots, k\}$ started from singletons: while there are $b$ blocks, each given set of $r$ of them, $2 \le r \le b$, merges into one at rate $\int_0^1 x^{r-2} (1 - x)^{b-r} \, dx = \frac{(r-2)!\,(b-r)!}{(b-1)!}$.
Source: Brunet, Derrida, Mueller and Munier (2006, 2007), for branching–selection models of N particles; for the N-BBM, J. Berestycki (2014), Conjecture 86. Stated by AIM — Open Applied Problems in this stationary, finite-window form.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, branching Brownian motion, coalescents, genealogies
References
@article{brunet2006,
author = {Brunet, {\'E}ric and Derrida, Bernard and Mueller, Alfred H. and Munier, St{\'e}phane},
title = {Noisy traveling waves: effect of selection on genealogies},
journal = {Europhys. Lett.},
volume = {76},
number = {1},
year = {2006},
pages = {1--7},
doi = {10.1209/epl/i2006-10224-4},
}
@article{brunet2007,
author = {Brunet, {\'E}ric and Derrida, Bernard and Mueller, Alfred H. and Munier, St{\'e}phane},
title = {Effect of selection on ancestry: an exactly soluble case and its phenomenological generalization},
journal = {Phys. Rev. E},
volume = {76},
number = {4},
year = {2007},
pages = {041104},
doi = {10.1103/PhysRevE.76.041104},
}
@misc{berestycki2014,
author = {Berestycki, Julien},
title = {Topics on branching {B}rownian motion},
howpublished = {Lecture notes, EBP XVIII},
year = {2014},
url = {https://www.stats.ox.ac.uk/~berestyc/Articles/EBP18_v2.pdf},
note = {Version of 7 August 2014; Section 9.1.3, Conjecture 86},
}
@misc{berestycki2024,
author = {Berestycki, Julien and Tough, Oliver},
title = {Selection principle for the {$N$}-{BBM}},
year = {2024},
eprint = {2407.05792},
archivePrefix = {arXiv},
}
@article{berestycki2013,
author = {Berestycki, Julien and Berestycki, Nathana{\"e}l and Schweinsberg, Jason},
title = {The genealogy of branching {B}rownian motion with absorption},
journal = {Ann. Probab.},
volume = {41},
number = {2},
year = {2013},
pages = {527--618},
doi = {10.1214/11-AOP728},
eprint = {1001.2337},
archivePrefix = {arXiv},
}
@article{maillard2016,
author = {Maillard, Pascal},
title = {Speed and fluctuations of {$N$}-particle branching {B}rownian motion with spatial selection},
journal = {Probab. Theory Related Fields},
volume = {166},
number = {3-4},
year = {2016},
pages = {1061--1173},
doi = {10.1007/s00440-016-0701-9},
eprint = {1304.0562},
archivePrefix = {arXiv},
}
@misc{aim-selected-brownian-population-genealogy,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Bolthausen–Sznitman ancestry for a fixed-size selected population}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/318-selected-brownian-population-genealogy.md},
note = {Entry 318 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00471, "Bolthausen–Sznitman ancestry for a fixed-size selected population", clopen.solutions.