有限图上的点强化跳跃过程和随机Schr dinger算子
The Vertex Reinforced Jump Process and a Random Schrödinger operator on finite graphs
摘要：
我们引入了一个新的指数族的概率分布，它可以被看作是一个多元推广的逆高斯分布。作为随机薛定谔算子的势，这个指数族与给出顶点强化跳跃过程（VRJP）的混合测度的随机场有关，因此与边强化随机游动（ERRW）的混合测度有关，即所谓的魔术公式。特别是，它通过直接计算这些混合措施，解决了Diaconis提出的一个问题的归一化常数的值。本文的结果在[Sabot-Zeng，2015]中是有用的，其中证明了VRJP和ERRW的几个性质，特别是瞬态区域中的泛函中心极限定理，以及二维ERRW的递归性。
英文摘要：
We introduce a new exponential family of probability distributions, which can be viewed as a multivariate generalization of the Inverse Gaussian distribution. Considered as the potential of a random Schrödinger operator, this exponential family is related to the random field that gives the mixing measure of the Vertex Reinforced Jump Process (VRJP), and hence to the mixing measure of the Edge Reinforced Random Walk (ERRW), the so-called magic formula. In particular, it yields by direct computation the value of the normalizing constants of these mixing measures, which solves a question raised by Diaconis. The results of this paper are instrumental in [Sabot-Zeng,2015], where several properties of the VRJP and the ERRW are proved, in particular a functional central limit theorem in transient regimes, and recurrence of the 2-dimensional ERRW.
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