ترغب بنشر مسار تعليمي؟ اضغط هنا

Reversed Dirichlet environment and directional transience of random walks in Dirichlet random environment

124   0   0.0 ( 0 )
 نشر من قبل Laurent Tournier
 تاريخ النشر 2009
  مجال البحث
والبحث باللغة English
 تأليف Christophe Sabot




اسأل ChatGPT حول البحث

We consider random walks in a random environment that is given by i.i.d. Dirichlet distributions at each vertex of Z^d or, equivalently, oriented edge reinforced random walks on Z^d. The parameters of the distribution are a 2d-uplet of positive real numbers indexed by the unit vectors of Z^d. We prove that, as soon as these weights are nonsymmetric, the random walk in this random environment is transient in a direction with positive probability. In dimension 2, this result can be strenghened to an almost sure directional transience thanks to the 0-1 law from [ZM01]. Our proof relies on the property of stability of Dirichlet environment by time reversal proved in [Sa09]. In a first part of this paper, we also give a probabilistic proof of this property as an alternative to the change of variable computation used in that article.



قيم البحث

اقرأ أيضاً

117 - Christophe Sabot 2010
We consider random walks in random Dirichlet environment (RWDE) which is a special type of random walks in random environment where the exit probabilities at each site are i.i.d. Dirichlet random variables. On $Z^d$, RWDE are parameterized by a $2d$- uplet of positive reals. We prove that for all values of the parameters, RWDE are transient in dimension $dge 3$. We also prove that the Green function has some finite moments and we characterize the finite moments. Our result is more general and applies for example to finitely generated symmetric transient Cayley graphs. In terms of reinforced random walks it implies that directed edge reinforced random walks are transient for $dge 3$.
158 - Christophe Sabot 2010
We consider random walks in random Dirichlet environment (RWDE) which is a special type of random walks in random environment where the exit probabilities at each site are i.i.d. Dirichlet random variables. On ${mathbb Z}^d$, RWDE are parameterized b y a 2d-uplet of positive reals called weights. In this paper, we characterize for $dge 3$ the weights for which there exists an absolutely continuous invariant probability for the process viewed from the particle. We can deduce from this result and from [27] a complete description of the ballistic regime for $dge 3$.
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 st rong 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.
162 - Yueyun Hu , Nobuo Yoshida 2007
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 gro ws 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.
The integer points (sites) of the real line are marked by the positions of a standard random walk. We say that the set of marked sites is weakly, moderately or strongly sparse depending on whether the jumps of the standard random walk are supported b y a bounded set, have finite or infinite mean, respectively. Focussing on the case of strong sparsity we consider a nearest neighbor random walk on the set of integers having jumps $pm 1$ with probability $1/2$ at every nonmarked site, whereas a random drift is imposed at every marked site. We prove new distributional limit theorems for the so defined random walk in a strongly sparse random environment, thereby complementing results obtained recently in Buraczewski et al. (2018+) for the case of moderate sparsity and in Matzavinos et al. (2016) for the case of weak sparsity. While the random walk in a strongly sparse random environment exhibits either the diffusive scaling inherent to a simple symmetric random walk or a wide range of subdiffusive scalings, the corresponding limit distributions are non-stable.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا