Bertacchi, Daniela; Coletti, Cristian F.; Zucca, Fabio Global survival of branching random walks and tree-like branching random walks. (English) Zbl 1364.60102 ALEA, Lat. Am. J. Probab. Math. Stat. 14, No. 1, 381-402 (2017). Summary: The reproduction speed of a continuous-time branching random walk is proportional to a positive parameter \(\lambda\). There is a threshold for \(\lambda\), which is called \(\lambda_w\), that separates almost sure global extinction from global survival. Analogously, there exists another threshold \(\lambda_s\) below which any site is visited almost surely a finite number of times (i.e. local extinction) while above it there is a positive probability of visiting every site infinitely many times. The local critical parameter \(\lambda_s\) is completely understood and can be computed as a function of the reproduction rates. On the other hand, only for some classes of branching random walks it is known that the global critical parameter \(\lambda_w\) is the inverse of a certain function of the reproduction rates, which we denote by \(K_w\). We provide here new sufficient conditions which guarantee that the global critical parameter equals \(1/K_w\). This result extends previously known results for branching random walks on multigraphs and general branching random walks. We show that these sufficient conditions are satisfied by periodic tree-like branching random walks. We also discuss the critical parameter and the critical behaviour of continuous-time branching processes in varying environment. So far, only examples where \(\lambda_w= 1/K_w\) were known; here we provide an example where \(\lambda_w > 1/K_w\). Cited in 1 Document MSC: 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60K35 Interacting random processes; statistical mechanics type models; percolation theory Keywords:branching random walk; branching process; local survival; global survival; varying environment; tree-like; critical parameters; generating function × Cite Format Result Cite Review PDF Full Text: arXiv Link References: [1] A. B. Adcock, B. D. Sullivan and M. W. Mahoney. Tree-like structure in largesocial and information networks. {\it Proc. IEEE ICDM }pages 1-10 (2013).DOI:10.1109/ICDM.2013.77 [2] D. Bertacchi, P. M. Rodriguez and F. Zucca. Galton-watson processes in varyingenvironment and accessibility percolation. {\it ArXiv Mathematics e-prints }(2016) [3] D. Bertacchi and F. Zucca. Critical behaviors and critical values of branching ran-dom walks on multigraphs. {\it J. Appl. Probab. }45 (2), 481-497 (2008) · Zbl 1144.60057 [4] D. Bertacchi and F. Zucca. 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