随机划分与量子Benjamin-Ono族
Random Partitions and the Quantum Benjamin-Ono Hierarchy
摘要：
我们从应用于经典周期Benjamin-Ono方程临界正则性$s= -1/2$的相干态半经典分析的一般结果中获得随机分区的精确和渐近结果。我们发现了这个系统的经典$dF_{\star |v} (c| \overline{\varepsilon})$和量子$d\hat{F}^{η_{NS}}( c | \hbar, \overline{\varepsilon})|_Ψ$守恒密度，色散系数$\overline{\varepsilon}$扩展了Nazarov-Sklyanin（2013）。对于量子定态，这个守恒密度是$dF_λ(c | \varepsilon_2, \varepsilon_1)$，即$\hbar = - \varepsilon_1 \varepsilon_2$的各向异性$(\varepsilon_2, \varepsilon_1) \in \mathbb{C}^2$的分区$λ$的轮廓的瑞利测度，$\overline{\varepsilon}= \varepsilon_1 + \varepsilon_2$在$\varepsilon_2 \longleftrightarrow \varepsilon_1$下不变。由于Jack多项式是量子定态，Stanley的Cauchy核（1989）是再生核，相干态$Υ_v ( \cdot | \hbar)$中量子周期Benjamin-Ono族的随机值是划分上的“Jack测度”，是Okounkov的Schur测度（1999）的色散推广。通过我们对相干态的一般结果，我们得到了$\hbar \rightarrow 0$的极限形状，即$v$的经典守恒密度，量子涨落是一个显式的高斯场。我们的研究结果如下从一个枚举渐近展开$\hbar$和$\overline{\varepsilon}$的联合累积量在新的组合对象，我们称之为“带状路径”。我们的结果反映了这样一个事实：在固定的$\hbar>0$下，定义Fock空间的权已经是方差分数布朗运动$\hbar$和Hurst指数$(-s) - \tfrac{1}{2} \dim \mathbb{T} = + \tfrac{1}{2} - \tfrac{1}{2} = 0.$
英文摘要：
We derive exact and asymptotic results for random partitions from general results in the semi-classical analysis of coherent states applied to the classical periodic Benjamin-Ono equation at critical regularity $s= -1/2$. We find classical $dF_{\star |v} (c| \overline{\varepsilon})$ and quantum $d\hat{F}^{η_{NS}}( c | \hbar, \overline{\varepsilon})|_Ψ$ conserved densities for this system with dispersion coefficient $\overline{\varepsilon}$ extending Nazarov-Sklyanin (2013). For quantum stationary states, this conserved density is $dF_λ(c | \varepsilon_2, \varepsilon_1)$ the Rayleigh measure of the profile of a partition $λ$ of anisotropy $(\varepsilon_2, \varepsilon_1) \in \mathbb{C}^2$ for $\hbar = - \varepsilon_1 \varepsilon_2$, $\overline{\varepsilon}= \varepsilon_1 + \varepsilon_2$ invariant under $\varepsilon_2 \longleftrightarrow \varepsilon_1$. As Jack polynomials are the quantum stationary states and Stanley's Cauchy kernel (1989) is the reproducing kernel, the random values of the quantum periodic Benjamin-Ono hierarchy in a coherent state $Υ_v ( \cdot | \hbar)$ are a "Jack measure" on partitions, a dispersive generalization of Okounkov's Schur measures (1999). By our general results for coherent states, we have concentration on a limit shape as $\hbar \rightarrow 0$, the classical conserved density at $v$, and quantum fluctuations are an explicit Gaussian field. Our results follow from an enumerative asymptotic expansion in $\hbar$ and $\overline{\varepsilon}$ of joint cumulants over new combinatorial objects we call "ribbon paths". Our results reflect the fact that at fixed $\hbar>0$ the weight defining Fock space is already a fractional Brownian motion of variance $\hbar$ and Hurst index $(-s) - \tfrac{1}{2} \dim \mathbb{T} = + \tfrac{1}{2} - \tfrac{1}{2} = 0.$
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。
