2017/03/13 by Daniela Bertacchi, Bertacchi, Daniela, Cristian F. Coletti +3 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #60J80 #60K35 #Bayesian Methods and Mixture Models #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1703.04499
openalex publication_date 2017/03/13 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28
The reproduction speed of a continuous-time branching random walk is\nproportional to a positive parameter \λ. There is a threshold for\n\λ, which is called \λw, that separates almost sure global\nextinction from global survival. Analogously, there exists another threshold\n\λs below which any site is visited almost surely a finite number of\ntimes (i.e.~local extinction) while above it there is a positive probability of\nvisiting every site infinitely many times. The local critical parameter\n\λs is completely understood and can be computed as a function of the\nreproduction rates. On the other hand, only for some classes of branching\nrandom walks it is known that the global critical parameter \λw is the\ninverse of a certain function of the reproduction rates, which we denote by\nKw. We provide here new sufficient conditions which guarantee that the\nglobal critical parameter equals 1/Kw. This result extends previously known\nresults for branching random walks on multigraphs and general branching random\nwalks. We show that these sufficient conditions are satisfied by periodic\ntree-like branching random walks. We also discuss the critical parameter and\nthe critical behaviour of continuous-time branching processes in varying\nenvironment. So far, only examples where \λw=1/Kw were known; here we\nprovide an example where \λw>1/Kw.\n