Time-inhomogeneous N-particle Branching Brownian Motion and the continuous random energy model

[2402.04917] Time-inhomogeneous N-particle Branching Brownian Motion and the continuous random energy model Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Probability arXiv:2402.04917 (math) [Submitted on 7 Feb 2024 ( v1 ), last revised 9 Apr 2026 (this version, v4)] Title: Time-inhomogeneous N-particle Branching Brownian Motion and the continuous random energy model Authors: Alexandre Legrand , Pascal Maillard View a PDF of the paper titled Time-inhomogeneous N-particle Branching Brownian Motion and the continuous random energy model, by Alexandre Legrand and Pascal Maillard View PDF HTML (experimental) Abstract: The $N$-particle branching Brownian motion ($N$-BBM) is a branching Markov process which describes the evolution of a population of particles undergoing reproduction and selection. It has attracted a lot of interest due to its relations to the study of front propagation phenomena on the one hand, and to (hierarchical) physical $p$-spin models on the other hand, among which the continuous random energy model (CREM). This paper investigates the asymptotic displacement of the $N$-BBM in a time-inhomogeneous setting, and when the time horizon $T$ and the number of particles $N$ jointly tend to infinity. We estimate the maximal displacement of the process up to the second order, and show that the latter undergoes a transition at the scale $\log N\approx T^{1/3}$. In particular when $\log N\ll T^{1/3}$ we recover the Brunet-Derrida behavior which was proven in a time-homogeneous setting and for $T\to+\infty$ then $N\to+\infty$. Furthermore, our results can also be interpreted from the perspective of algorithmic optimisation on some spin glass models, since the time-inhomogeneous $N$-BBM can be seen as the realization of an optimization procedure called beam search on the aforementioned CREM. The CREM has been proven by L. Addario-Berry and the second author to undergo an algorithm hardness threshold phenomenon, and the results of the present paper describe precisely the efficiency of the beam search algorithm around that threshold. Comments: 76 pages, 5 figures Subjects: Probability (math.PR) MSC classes: 60J80, 68Q17, 82C21 (Primary) 60J70, 92D25, 60K35 (Secondary) Cite as: arXiv:2402.04917 [math.PR] (or arXiv:2402.04917v4 [math.PR] for this version) https://doi.org/10.48550/arXiv.2402.04917 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Alexandre Legrand [ view email ] [v1] Wed, 7 Feb 2024 14:43:48 UTC (383 KB) [v2] Wed, 11 Sep 2024 16:36:27 UTC (389 KB) [v3] Fri, 13 Dec 2024 13:24:47 UTC (736 KB) [v4] Thu, 9 Apr 2026 10:14:57 UTC (731 KB) Full-text links: Access Paper: View a PDF of the paper titled Time-inhomogeneous N-particle Branching Brownian Motion and the continuous random energy model, by Alexandre Legrand and Pascal Maillard View PDF HTML (experimental) TeX Source view license Current browse context: math.PR < prev | next > new | recent | 2024-02 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... 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