Upper moderate deviation probabilities for the maximum of a branching random walk

Consider $M_n$ the maximal position at generation $n$ of a supercritical branching random walk. A\"id\'ekon (2013) obtained and described the convergence in law, as time $n$ goes to infinity, of $M_n-m_n$, where $m_n$ is an explicit function. Equivalently, he identified the limit of $\mathbb{P}(M_n > m_n + x)$, for any $x \in \mathbb{R}$. More recently, Luo (2025) gave an asymptotic equivalent for the upper large deviation probability, that is $\mathbb{P}(M_n > m_n + xn)$, for $x > 0$. In this work, we study an intermediate regime, called upper moderate deviation. We obtain, under close-to-optimal integrability conditions, an asymptotic equivalent for $\mathbb{P}(M_n > m_n + x_n)$, where $x_n$ is such that $x_n \to \infty$ and $x_n = O(\sqrt{n})$. Our proof is based on a strategy due to Bramson, Ding, and Zeitouni (2016). As a byproduct, we obtain information about the typical behavior of particles contributing to such deviations. Finally, we apply our main result to show the convergence in law of the centered maximum of a two-speed branching random walk in the mean regime and describe its limit.