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OP-8062Status unassessedAnalysis

Riemann-Hilbert Problems · Question 7

Riemann-Hilbert problems, Toeplitz matrices, and applications · Historical workshop question

Historical list · status unassessedAs of 2026-09-08Source & evidence ↓
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Problem statement

Consider

ZN=∫Γ⋯∫Γ∏i≤j<k≤N∣zk−zj∣2β∏j=1N(11+∣zj∣2)(N−1)β+1d∣z1∣⋯d∣zN∣.Z_N=\int_{\Gamma} \cdots \int_{\Gamma} \prod_{i \leq j < k \leq N} |z_k-z_j|^{2\beta} \prod_{j=1}^N\Big(\frac{1}{1+|z_j|^2}\Big)^{(N-1)\beta+1} d|z_1| \cdots d|z_N|.

Study ZNZ_N as N→∞N \rightarrow \infty, where Γ\Gamma is a smooth loop on the Riemann sphere.

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References
  1. [1]Riemann-Hilbert problems, Toeplitz matrices, and applications link

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