标题： 退火几乎周期熵 显示英文标题

标题： Annealed almost periodic entropy

摘要： 这项工作研究了可以与以下内容相关联的熵的一些概念：(i) 分离的、单位的C*-代数$\mathfrak{A}$的表示，以及 (ii) 有限维表示$\mathfrak{A}$的辅助随机序列$(\pi_n)_{n\ge 1}$。 这延续了之前对这些熵概念性质的研究，当每个$\pi_n$是确定性的时，发现了与遍历理论中的熵以及非交换的Szegő极限定理推广之间的多种类比。 我们为上述 (i) 和 (ii) 中的数据关联了两种新的熵的概念： “退火”AP熵，这大致是确定性AP熵的一阶矩平均；以及“零阶”AP熵，它控制某些正定函数在表示$\pi_n$中出现的大偏差概率。 在发展了一些这方面的普遍理论之后，我们接着关注一个特殊情况，其中$\mathfrak{A}$是一个有限生成自由群的 C*-代数，而每个$\pi_n$是通过从 Haar 测度中独立随机选择一个由$n$-by-$n$么正矩阵组成的元组来生成的。 在这种情况下，可以为我们的某些熵的概念推导出显式公式，并由此获得随机矩阵理论中的新的大偏差原理。 显示英文摘要

摘要： This work studies certain notions of entropy that can be associated to (i) a representation of a separable, unital C*-algebra $\mathfrak{A}$ and (ii) an auxiliary random sequence $(\pi_n)_{n\ge 1}$ of finite-dimensional representations of $\mathfrak{A}$. This continues a previous research program into the properties of these entropy notions when each $\pi_n$ is deterministic, which uncovered a range of analogies with entropy in ergodic theory and also with non-commutative generalizations of Szeg\H{o}'s limit theorems. We associate two new notions of entropy to data as in (i) and (ii) above: `annealed' AP entropy, which is roughly a kind of first-moment average of deterministic AP entropies; and `zeroth-order' AP entropy, which controls the large deviations probabilities that certain positive definite functions appear in the representations $\pi_n$ at all. After developing some of this general theory, we then focus on the special case in which $\mathfrak{A}$ is the group C*-algebra of a finitely-generated free group and each $\pi_n$ is generated by choosing a tuple of $n$-by-$n$ unitary matrices independently at random from Haar measure. In that case, explicit formulas can be derived for some of our notions of entropy, and new large deviations principles in random matrix theory are obtained as a consequence.