$q$-形变 Perelomov-Popov 测度与量子化自由概率
$q$-deformed Perelomov-Popov measures and quantized free probability
AI总结：
本文引入依赖参数 $q$ 的离散 Perelomov-Popov 测度，证明其大数定律并计算极限的自由与无穷小自由累积量，建立 $q=0,1$ 结果间的插值及 Markov-Krein 型关系。
摘要：
对随机非递增整数元组的渐近研究，对于来自渐近表示论和统计物理的概率模型至关重要。我们研究此类元组的全局行为，引入一族依赖于参数 $q\in[-1,1]$ 的新的离散概率测度。我们基于 Schur 生成函数的渐近性质证明了这些测度的大数定律，并给出了极限测度的矩与自由累积量的显式公式。我们的结果在 Bufetov 和 Gorin 关于 $q=0,1$ 的结果之间给出了插值；他们从自由概率论的角度区分了这两种情形。我们展示了与自由概率论的联系，并受自由累积量公式的启发，引入了一种自由卷积的形变。我们还研究了大数定律的一阶修正，并将其与无穷小自由概率建立联系，显式计算了无穷小矩和无穷小自由累积量。最后，我们证明了不同 $q$ 对应极限测度之间的非渐近关系，这些关系与著名的 Markov-Krein 对应相关。
英文摘要：
The asymptotic study of tuples of random non-increasing integers is crucial for probabilistic models coming from asymptotic representation theory and statistical physics. We study the global behavior of such tuples, introducing a new family of discrete probability measures, depending on a parameter $q \in [- 1, 1]$. We prove the Law of Large Numbers for these measures based on the asymptotics of the Schur generating functions and we provide explicit formulas for the moments and the free cumulants of the limiting measures. Our results provide an interpolation between the results of Bufetov and Gorin for $q = 0, 1$, who distinguished these two cases from the side of free probability theory. We show the connection with free probability theory and we introduce a deformation of free convolution, motivated by our formulas for the free cumulants. We also study the first order correction to the Law of Large Numbers and we make the connection with infinitesimal free probability, computing explicitly the infinitesimal moments and the infinitesimal free cumulants. Finally, we prove non-asymptotic relations between the limiting measures for different $q$, which are related to the celebrated Markov-Krein correspondence.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
