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Intermittency for branching random walk in Pareto environment. (English) Zbl 1352.60133
The authors consider a branching random walk on an integer lattice \(\mathbb Z^d\), where the branching rates are given by i.i.d. Pareto random variables. At time \(0\), start with a single particle at the origin. The particle performs a continuous-time, nearest neighbor symmetric random walk on \(\mathbb Z^d\). When located at \(z\in\mathbb Z^d\), the particle splits into two new particles at rate \(\xi(z)\), where \(\{\xi(z): z\in\mathbb Z^d\}\) are i.i.d. random variables with \(\mathbb P[\xi(z) > x] = x^{-\alpha}\), \(x\geq 1\). The new particles behave in the same way as the original one. Let \(N(z,t)\) be the number of particles at site \(z\in\mathbb Z^d\) and time \(t\geq 0\). The main result obtained by the authors states that the process \[ M_T(z,t) := \frac{1}{a(T) T} \log_+ N([r(T) z], tT), \quad z\in\mathbb R^d, \; t\geq 0, \] with \[ a(T) = \left(\frac T {\log T}\right)^{\frac{d}{\alpha-d}}, \quad r(T) = \left(\frac T {\log T}\right)^{\frac{\alpha}{\alpha-d}}, \] can be approximated, for large \(T\), by a certain process \(m_T(z,t)\) which has an elegant description in terms of growing “lilypads”. From this result, the authors derive a number of consequences including an asymptotic description of the support of the branching random walk.

MSC:
60K37 Processes in random environments
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60G50 Sums of independent random variables; random walks
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