On the maximal displacement of critical branching random walk in random environment
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909545277685760 |
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| author | Fu, Wenxin Hong, Wenming |
| author_facet | Fu, Wenxin Hong, Wenming |
| contents | In this article, we study the maximal displacement of critical branching random walk in random environment. Let $M_n$ be the maximal displacement of a particle in generation $n$, and $Z_n$ be the total population in generation $n$, $M$ be the rightmost point ever reached by the branching random walk.
Under some reasonable conditions, we prove a conditional limit theorem,
\begin{equation*}
\mathcal{L}\left( \dfrac{M_n}{\sqrtσ n^{\frac{3}{4}}} |Z_n>0\right) \dcon \mathcal{L}\left(A_Λ\right),
\end{equation*}
where
random variable $A_Λ$ is related to the standard Brownian meander. And there exist some positive constant $C_1$ and $C_2$, such that
\begin{equation*}
C_1\leqslant\liminf\limits_{x\rightarrow\infty}x^{\frac{2}{3}}¶(M>x) \leqslant
\limsup\limits_{x\rightarrow\infty} x^{\frac{2}{3}}¶(M>x) \leqslant C_2.
\end{equation*}
Compared with the constant environment case (Lalley and Shao (2015)),
it revaels that, the conditional limit speed for $M_n$ in random environment (i.e., $n^{\frac{3}{4}}$) is significantly greater than that of constant environment case (i.e., $n^{\frac{1}{2}}$), and so is the tail probability for the $M$ (i.e., $x^{-\frac{2}{3}}$ vs $x^{-2}$).
Our method is based on the path large deviation for the reduced critical branching random walk in random environment. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_15841 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the maximal displacement of critical branching random walk in random environment Fu, Wenxin Hong, Wenming Probability In this article, we study the maximal displacement of critical branching random walk in random environment. Let $M_n$ be the maximal displacement of a particle in generation $n$, and $Z_n$ be the total population in generation $n$, $M$ be the rightmost point ever reached by the branching random walk. Under some reasonable conditions, we prove a conditional limit theorem, \begin{equation*} \mathcal{L}\left( \dfrac{M_n}{\sqrtσ n^{\frac{3}{4}}} |Z_n>0\right) \dcon \mathcal{L}\left(A_Λ\right), \end{equation*} where random variable $A_Λ$ is related to the standard Brownian meander. And there exist some positive constant $C_1$ and $C_2$, such that \begin{equation*} C_1\leqslant\liminf\limits_{x\rightarrow\infty}x^{\frac{2}{3}}¶(M>x) \leqslant \limsup\limits_{x\rightarrow\infty} x^{\frac{2}{3}}¶(M>x) \leqslant C_2. \end{equation*} Compared with the constant environment case (Lalley and Shao (2015)), it revaels that, the conditional limit speed for $M_n$ in random environment (i.e., $n^{\frac{3}{4}}$) is significantly greater than that of constant environment case (i.e., $n^{\frac{1}{2}}$), and so is the tail probability for the $M$ (i.e., $x^{-\frac{2}{3}}$ vs $x^{-2}$). Our method is based on the path large deviation for the reduced critical branching random walk in random environment. |
| title | On the maximal displacement of critical branching random walk in random environment |
| topic | Probability |
| url | https://arxiv.org/abs/2503.15841 |