Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.20833 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908903626768384 |
|---|---|
| author | Flath, Gabriel |
| author_facet | Flath, Gabriel |
| contents | Consider a branching Brownian motion (BBM). It is well known \cite{Bramson1983ConvergenceOS, Lalley1987ACL} that the rightmost particle is located near \( m_t = \sqrt{2} t - \frac{3}{2\sqrt{2}} \log t \). Let $\mathcal{N}(t,x)$ be the set of particles within distance $x$ from $m_t$, where $x = o(t)$ grows with $t$. We prove that \(\#\mathcal{N}(t,x)/π^{-1/2}xe^{xm_t/t} e^{-x^2/(2t)} \) converges in probability to $Z_\infty$, the limit of the so-called derivative martingale, and that, for \( x = O( t^{1/3}) \), the convergence cannot be strengthened to an almost sure result. Moreover, we prove that the asymptotic overlap distribution of two particles sampled uniformly from $\mathcal{N}(t,x)$ converges to that of the critical derivative martingale measure. This establishes a universal genealogical picture of the BBM front at sublinear distances from the tip. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_20833 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The number of particles at sublinear distances from the tip in branching Brownian motion Flath, Gabriel Probability Consider a branching Brownian motion (BBM). It is well known \cite{Bramson1983ConvergenceOS, Lalley1987ACL} that the rightmost particle is located near \( m_t = \sqrt{2} t - \frac{3}{2\sqrt{2}} \log t \). Let $\mathcal{N}(t,x)$ be the set of particles within distance $x$ from $m_t$, where $x = o(t)$ grows with $t$. We prove that \(\#\mathcal{N}(t,x)/π^{-1/2}xe^{xm_t/t} e^{-x^2/(2t)} \) converges in probability to $Z_\infty$, the limit of the so-called derivative martingale, and that, for \( x = O( t^{1/3}) \), the convergence cannot be strengthened to an almost sure result. Moreover, we prove that the asymptotic overlap distribution of two particles sampled uniformly from $\mathcal{N}(t,x)$ converges to that of the critical derivative martingale measure. This establishes a universal genealogical picture of the BBM front at sublinear distances from the tip. |
| title | The number of particles at sublinear distances from the tip in branching Brownian motion |
| topic | Probability |
| url | https://arxiv.org/abs/2504.20833 |