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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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