• Corpus ID: 119123279

Random partitions and the quantum Benjamin-Ono hierarchy

@article{Moll2016RandomPA,
  title={Random partitions and the quantum Benjamin-Ono hierarchy},
  author={Alexander Moll},
  journal={arXiv: Mathematical Physics},
  year={2016}
}

• Alexander Moll
• Published 12 August 2015
• Mathematics
• arXiv: Mathematical Physics

Jack measures $M_V (\varepsilon_2, \varepsilon_1)$ on partitions $\lambda$ are discrete stochastic processes occurring naturally in the study of continuum circular $\beta$-ensembles in generic background potentials $V$ and arbitrary values $\beta$ of Dyson's inverse temperature. For analytic $V$, we prove a law of large numbers and central limit theorem in the scaling limit $\varepsilon_2 \rightarrow 0 \leftarrow \varepsilon_1$ taken at fixed inverse Jack parameter $\beta / 2=- \varepsilon_2 / \varepsilon_1>0$. We characterize the limit shape and macroscopic fluctuations as push-forwards along $v: \mathbb{T} \rightarrow \mathbb{R}$ of the uniform measure on $\mathbb{T}$ (LLN) and of the restriction to $\mathbb{T} \cap \mathbb{H}_+$ of a Gaussian free field on the upper half-plane (CLT), respectively, where $v = \Delta^{1/2} V$. At $\beta=2$, this recovers Okounkov's LLN for Schur measures (2005) and coincides with Breuer-Duits' CLT for biorthogonal ensembles (2013).
Our limit theorems follow from an all-order expansion of the joint cumulants of the linear statistics of ${M}_V(\varepsilon_2, \varepsilon_1)$ (AOE). This result has the same form as the all-order $1/N$ refined topological expansion for the $\beta$-ensembles on the line in a one-cut potential $V$ at inverse temperature $\beta$ obtained for formal $V$ by Chekhov-Eynard (2006) and for convergent $V$ by Borot-Guionnet (2012). To prove our AOE, we rely on the Lax operator for the quantum Benjamin-Ono hierarchy with periodic profile $v$ exhibited in collective field variables by Nazarov-Sklyanin (2013). The characterization of the limiting laws as push-forwards follows from factorization formulas for resolvents of Toeplitz operators on $\mathbb{T}$ with symbol $v$ due to Krein and Calder\'{o}n-Spitzer-Widom (1958).