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Register a new userAging and quenched localization for one-dimensional random walks in random environment in the sub-ballistic regime
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Authors
Nathanael Enriquez
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We consider transient one-dimensional random walks in random environment with zero asymptotic speed. An aging phenomenon involving the generalized Arcsine law is proved using the localization of the walk at the foot of valleys of height $log t$. In the quenched setting, we also sharply estimate the distribution of the walk at time $t$.
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We consider branching random walks in $d$-dimensional integer lattice with time-space i.i.d. offspring distributions. This model is known to exhibit a phase transition: If $d ge 3$ and the environment is not too random, then, the total population grows as fast as its expectation with strictly positive probability. If,on the other hand, $d le 2$, or the environment is ``random enough, then the total population grows strictly slower than its expectation almost surely. We show the equivalence between the slow population growth and a natural localization property in terms of replica overlap. We also prove a certain stronger localization property, whenever the total population grows strictly slower than its expectation almost surely.
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We consider branching random walks in $d$-dimensional integer lattice with time-space i.i.d. offspring distributions. When $d ge 3$ and the fluctuation of the environment is well moderated by the random walk, we prove a central limit theorem for the density of the population, together with upper bounds for the density of the most populated site and the replica overlap. We also discuss the phase transition of this model in connection with directed polymers in random environment.
We give new criteria for ballistic behavior of random walks in random environment which are perturbations of the simple symmetric random walk on $mathbb Z^d$ in dimensions $dge 4$. Our results extend those of Sznitman [Ann. Probab. 31, no. 1, 285-322 (2003)] and the recent ones of Ramirez and Saglietti [Preprint, arXiv:1808.01523], and allow us to exhibit new examples in dimensions $dge 4$ of ballistic random walks which do not satisfy Kalikows condition. Our criteria implies ballisticity whenever the average of the local drift of the walk is not too small compared with an appropriate moment of the centered environment. The proof relies on a concentration inequality of Boucheron et al. [Ann. Probab. 33, no. 2, 514-560 (2005)].
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