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Register a new userSlowdown estimates for one-dimensional random walks in random environment with holding times
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We consider a one dimensional random walk in random environment that is uniformly biased to one direction. In addition to the transition probability, the jump rate of the random walk is assumed to be spatially inhomogeneous and random. We study the probability that the random walk travels slower than its typical speed and determine its decay rate asymptotic.
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We consider one-dimensional random walks in random environment which are transient to the right. Our main interest is in the study of the sub-ballistic regime, where at time $n$ the particle is typically at a distance of order $O(n^kappa)$ from the origin, $kappain(0,1)$. We investigate the probabilities of moderate deviations from this behaviour. Specifically, we are interested in quenched and annealed probabilities of slowdown (at time $n$, the particle is at a distance of order $O(n^{ u_0})$ from the origin, $ u_0in (0,kappa)$), and speedup (at time $n$, the particle is at a distance of order $n^{ u_1}$ from the origin, $ u_1in (kappa,1)$), for the current location of the particle and for the hitting times. Also, we study probabilities of backtracking: at time $n$, the particle is located around $(-n^ u)$, thus making an unusual excursion to the left. For the slowdown, our results are valid in the ballistic case as well.
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 $t$.
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.
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)].
We study survival of nearest-neighbour branching random walks in random environment (BRWRE) on ${mathbb Z}$. A priori there are three different regimes of survival: global survival, local survival, and strong local survival. We show that local and strong local survival regimes coincide for BRWRE and that they can be characterized with the spectral radius of the first moment matrix of the process. These results are generalizations of the classification of BRWRE in recurrent and transient regimes. Our main result is a characterization of global survival that is given in terms of Lyapunov exponents of an infinite product of i.i.d. $2times 2$ random matrices.
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