论文标题
关于与顶点增强了树木和Z D的顶点增强跳跃过程的Martingale的渐近行为
About the asymptotic behaviour of the martingale associated with the Vertex Reinforced Jump Process on trees and Z d
论文作者
论文摘要
我们研究了Martingale（$ψ$ n（o））n $ \ in $ n与顶点增强跳跃过程（VRJP）相关的渐近行为。我们表明，在VRJP的所有瞬态阶段，在树上的每个p> 1的L P中都限制在L p中。此外，当VRJP在树上复发时，我们对$ψ$ n（o）的矩进行了良好的估计，我们可以计算确切的降低速率$τ$，因此n -1 ln（$ψ$ n（o）$ \ sim $ - $τ$几乎与$τ$相关的位置与标准的随机步行相关的地方，此外，在树上，在关键点，我们表明n -1/3 ln（$ψ$ n（o））$ \ sim $ - $ρ$ c几乎可以肯定可以明确计算$ρ$ c。此外，在临界点，我们证明与VRJP相关的离散过程是阳性复发链的混合物。我们的证明使用与VRJP相关的$β$ - 电位的属性，以及来自分支随机步行域的技术。
We study the asymptotic behaviour of the martingale ($ψ$ n (o)) n$\in$N associated with the Vertex Reinforced Jump Process (VRJP). We show that it is bounded in L p for every p > 1 on trees and uniformly integrable on Z d in all the transient phase of the VRJP. Moreover, when the VRJP is recurrent on trees, we have good estimates on the moments of $ψ$ n (o) and we can compute the exact decreasing rate $τ$ such that n --1 ln($ψ$ n (o)) $\sim$ --$τ$ almost surely where $τ$ is related to standard quantities for branching random walks. Besides, on trees, at the critical point, we show that n --1/3 ln($ψ$ n (o)) $\sim$ --$ρ$ c almost surely where $ρ$ c can be computed explicitely. Furthermore, at the critical point, we prove that the discrete process associated with the VRJP is a mixture of positive recurrent Markov chains. Our proofs use properties of the $β$-potential associated with the VRJP and techniques coming from the domain of branching random walks.
