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# Branching Brownian motion conditioned on small maximum

Abstract : For a standard binary branching Brownian motion on the real line, it is known that the typical position of the maximal position $M_t$ among all particles alive at time $t$ is $m_t+\Theta(1)$ with $m_t=\sqrt{2}t-\frac{3}{2\sqrt{2}}\log t$. Further, it is proved independently in \cite{ABBS13} and \cite{ABK13} that the branching Brownian motion shifted by $m_t$ (or $M_t$) converges in law to some decorated Poisson point process. The goal of this work is to study the branching Brownian motion conditioned on $M_t\ll m_t$. We give a complete description of the limiting extremal process conditioned on $\{M_t\le \sqrt{2}\alpha t\}$ with $\alpha<1$, which reveals a phase transition at $\alpha=1-\sqrt{2}$. We also verify the conjecture of Derrida and Shi \cite{DS} on the precise asymptotic behaviour of $\P(M_t \leq \sqrt{2}\alpha t)$ for $\alpha < 1$.
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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-02887692
Contributor : Bastien Mallein Connect in order to contact the contributor
Submitted on : Friday, November 18, 2022 - 8:22:56 AM
Last modification on : Thursday, November 24, 2022 - 3:32:48 AM

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manuscript6.pdf
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### Identifiers

• HAL Id : hal-02887692, version 1
• ARXIV : 2007.00405

### Citation

Xinxin Chen, Hui He, Bastien Mallein. Branching Brownian motion conditioned on small maximum. 2022. ⟨hal-02887692⟩

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