### Abstract

We study the determinant $${\det(I-\gamma K_s), 0 < \gamma < 1}$$det(I-γKs),0<γ<1 , of the integrable Fredholm operator K_{s} acting on the interval (−1, 1) with kernel $${K_s(\lambda, \mu)= \frac{\sin s(\lambda - \mu)}{\pi (\lambda-\mu)}}$$Ks(λ,μ)=sins(λ-μ)π(λ-μ). This determinant arises in the analysis of a log-gas of interacting particles in the bulk-scaling limit, at inverse temperature $${\beta=2}$$β=2 , in the presence of an external potential $${v=-\frac{1}{2}\ln(1-\gamma)}$$v=-12ln(1-γ) supported on an interval of length $${\frac{2s}{\pi}}$$2sπ. We evaluate, in particular, the double scaling limit of $${\det(I-\gamma K_s)}$$det(I-γKs) as $${s\rightarrow\infty}$$s→∞ and $${\gamma\uparrow 1}$$γ↑1 , in the region $${0\leq\kappa=\frac{v}{s}=-\frac{1}{2s}\ln(1-\gamma)\leq 1-\delta}$$0≤κ=vs=-12sln(1-γ)≤1-δ , for any fixed $${0 < \delta < 1}$$0<δ<1. This problem was first considered by Dyson (Chen Ning Yang: A Great Physicist of the Twentieth Century. International Press, Cambridge, pp. 131–146, 1995).

Original language | English (US) |
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Pages (from-to) | 1397-1463 |

Number of pages | 67 |

Journal | Communications In Mathematical Physics |

Volume | 337 |

Issue number | 3 |

DOIs | |

State | Published - Aug 1 2015 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications In Mathematical Physics*,

*337*(3), 1397-1463. https://doi.org/10.1007/s00220-015-2357-1