无穷p-adic随机矩阵与p-adic华测度的遍历分解
Infinite p-adic random matrices and ergodic decomposition of p-adic Hua measures
摘要：
Neretin在无限$p$ -adic矩阵$Mat\left(\mathbb{N},\mathbb{Q}_p\right)$上构造了一个类似的华测度. Bufetov和Qiu对$Mat\left(\mathbb{N},\mathbb{Q}_p\right)$上在$GL(\infty,\mathbb{Z}_p)\times GL(\infty,\mathbb{Z}_p)$自然作用下不变的遍历测度进行了分类.本文解决了Neretin引入的$p$ -adic Hua测度的遍历分解问题.我们证明了支配遍历分解的概率测度有一个明确的表达式，它与划分上的Hall-Littlewood测度相一致。我们的论证涉及某些马尔可夫链。
英文摘要：
Neretin constructed an analogue of the Hua measures on the infinite $p$-adic matrices $Mat\left(\mathbb{N},\mathbb{Q}_p\right)$. Bufetov and Qiu classified the ergodic measures on $Mat\left(\mathbb{N},\mathbb{Q}_p\right)$ that are invariant under the natural action of $GL(\infty,\mathbb{Z}_p)\times GL(\infty,\mathbb{Z}_p)$. In this paper we solve the problem of ergodic decomposition for the $p$-adic Hua measures introduced by Neretin. We prove that the probability measure governing the ergodic decomposition has an explicit expression which identifies it with a Hall-Littlewood measure on partitions. Our arguments involve certain Markov chains.
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