On the maximal displacement of critical branching random walk in random environment

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}{\sqrt{\sigma} n^{\frac{3}{4}}} |Z_n>0\right) \dcon \mathcal{L}\left(A_\Lambda\right), \end{equation*} where random variable $A_\Lambda$ 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}}\P(M>x) \leqslant \limsup\limits_{x\rightarrow\infty} x^{\frac{2}{3}}\P(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.