Branching random walk in random environment phase transitions for local and global growth rates (Q1187107)

Let \((\eta_ n)\) be the infinite particle system on \(\mathbb{Z}\) whose evolution is as follows. At each unit of time each particle independently is replaced by a new generation. The size of a new generation descending from a particle at site \(x\) has distribution \(F_ x\) and each of its members independently jumps to site \(x\pm1\) with probability \((1\pm h)/2\), \(h\in[0,1]\). The sequence \(\{F_ x\}\) is i.i.d. with uniformly bounded second moment and is kept fixed during the evolution. The initial configuration \(\eta_ 0\) is shift invariant and ergodic. Two quantities are considered: (1) the global particle density \(D_ n\) (= large volume limit of number of particles per site at time \(n\)); (2) the local particle density \(d_ n\) (= average number of particles at site 0 at time \(n\)). We calculate the limits \(\rho\) and \(\lambda\) of \(n^{-1} \log(D_ n)\) and \(n^{-1} \log(d_ n)\) explicitly in the form of two variational formulas. Both limits (and variational formulas) do not depend on the realization of \(\{F_ x\}\) a.s. By analyzing the variational formulas we abstract how \(\rho\) and \(\lambda\) depend on the drift \(h\) for fixed distribution of \(F_ x\). It turns out that the system behaves in a way that is drastically different from what happens in a spatially homogeneous medium: (i) Both \(\rho(h)\) and \(\lambda(h)\) exhibit a phase transition associated with localization vs. delocalization at two respective critical values \(h_ 1\) and \(h_ 3\) in (0,1). Here the behavior of the path of descent of a typical particle in the whole population resp. in the population at 0 changes from moving on scale \(o(n)\) to moving on scale \(n\). We extract variational expressions for \(h_ 1\) and \(h_ 3\). (ii) Both \(\rho(h)\) and \(\lambda(h)\) change sign at two respective critical values \(h_ 2\) and \(h_ 4\) in (0,1) (for suitable distribution of \(F_ x\)). That is, the system changes from survival to extinction on a global resp. on a local scale.