The minimum of a branching random walk outside the boundary case (Q2405203)

For a supercritical branching random walk (BRW) on the line, denote by \(V_1\), \(V_2,\dots\) the positions of the first generation individuals. Set \(\psi(\beta):=\log \mathbb{E}\sum_{i\geq 1}e^{-\beta V_i}\) for \(\beta\in\mathbb{R}\). The standing assumptions of the present article are \(\psi(1)=1\), \(\psi^\prime(1-)<\infty\); the step distribution of an associated standard random walk belongs to the domain of attraction of a stable distribution with finite mean (which particularly entails \(\psi(1+)=\infty\)); some additional conditions which ensure, among others, that the Biggins martingale converges in mean. \par Denote by \(M_n\) the minimal position in the \(n\)th generation of the BRW. It is proved that \(M_n-\alpha_n\) converges in distribution to a random variable \(M\) with \(\mathbb{P}\{M>x\}=\mathbb{E}\exp(-ce^xW_\infty)\) for \(x\in\mathbb{R}\). Here, \(c>0\) is a constant, \(W_\infty\) is the limit of the Biggins martingale, and the centering constants \(\alpha_n\) are known explicitly. A special attention is given to the situation where \(V_1\), \(V_2,\ldots\) are i.i.d. which allows for some simplifications. The authors offer a nice intuitive reasoning behind their result. One section of the paper is devoted to interpreting the authors findings in the associated thermodynamics framework. \par In the principally different, so called boundary case \(\psi(1)=\psi^\prime(1-)=0\) \textit{E. Aïdékon} [Ann. Probab. 41, No. 3A, 1362--1426 (2013; Zbl 1285.60086)] has proved that the distribution of \(M_n\), centered with different constants, converges weakly to a distribution other than that of \(M\).