无限图上与顶点加强跳过程相关的随机Schr dinger算子
A random Schrödinger operator associated with the Vertex Reinforced Jump Process on infinite graphs
摘要：
本文研究了顶点强化跳跃过程（VRJP）和边强化随机游动（ERRW）及其与随机薛定谔算子的联系。在无限图上，我们定义了一个1依赖随机势$β$，扩展了[20]中在有限图上定义的随机势$β$，并考虑其相关的随机薛定谔算子$H_β$。我们构造了一个随机函数$ψ$作为鞅的极限，使得当VRJP是常返的时，$ψ$是$ψ=0$;当VRJP是瞬态的时，$ψ$是随机薛定谔算子的正的广义本征函数，其本征值为$0$.然后，我们证明了表示的VRJP无限图的马尔可夫跳跃过程的混合涉及的功能$ψ$，随机薛定谔算子的格林函数和一个独立的Gamma随机变量。在${\mathbb Z}^d$上，我们利用[10，8]的估计，从这个表示中推导出VRJP和ERRW的常返性或瞬变性的0 - 1定律，以及VRJP和ERRW在$d\ge 3$维弱强化处的泛函中心极限定理。最后，我们推导出了任意初始常权下ERRW在维数$ d=2$中的递推关系（使用Merkl和Rolles的估计，[15，17]），从而完整地回答了Diaconis的老问题。我们还提出了一些问题之间的联系复发/瞬态的VRJP和本地化/离域的随机薛定谔运营商$H_β$。
英文摘要：
This paper concerns the Vertex reinforced jump process (VRJP), the Edge reinforced random walk (ERRW) and their link with a random Schrödinger operator. On infinite graphs, we define a 1-dependent random potential $β$ extending that defined in [20] on finite graphs, and consider its associated random Schrödinger operator $H_β$. We construct a random function $ψ$ as a limit of martingales, such that $ψ=0$ when the VRJP is recurrent, and $ψ$ is a positive generalized eigenfunction of the random Schrödinger operator with eigenvalue $0$, when the VRJP is transient. Then we prove a representation of the VRJP on infinite graphs as a mixture of Markov jump processes involving the function $ψ$, the Green function of the random Schrödinger operator and an independent Gamma random variable. On ${\mathbb Z}^d$, we deduce from this representation a zero-one law for recurrence or transience of the VRJP and the ERRW, and a functional central limit theorem for the VRJP and the ERRW at weak reinforcement in dimension $d\ge 3$, using estimates of [10,8]. Finally, we deduce recurrence of the ERRW in dimension $ d=2$ for any initial constant weights (using the estimates of Merkl and Rolles, [15,17]), thus giving a full answer to the old question of Diaconis. We also raise some questions on the links between recurrence/transience of the VRJP and localization/delocalization of the random Schrödinger operator $H_β$.
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