Nonvanishing of \(\mathsf{X}_1(w)\) throughout the admissible complex-parameter range

Prove that the Hua–Pickrell random variable \(\mathsf{X}_1(w)\) is not almost surely zero for every complex parameter \(w\) satisfying \(\operatorname{Re}(2w)>-1\).

Background

The paper cites an unresolved question asking whether X1(w)\mathsf{X}_1(w) is nonzero with positive probability throughout the full admissible range Re⁡(2w)>−1\operatorname{Re}(2w)>-1. Before this work, the assertion was known for real ww with $2w>-1$.

Corollary \ref{almostsurely} proves the claim only when Re⁡(2w)>1\operatorname{Re}(2w)>1, by computing a strictly positive second moment. Thus the parameter region −1<Re⁡(2w)≤1-1<\operatorname{Re}(2w)\leq 1, including nonreal parameters, remains unresolved in the stated passage.

References

The above result thesecondmomentofXw provides a partial answer to the question posed in Remark 1.6, which asks whether $\mathsf{X}_{1}(w)$ is not almost surely zero in the full range $w \in \mathbb{C}$ with $\operatorname{Re}(2w) > -1$.

— Joint moments of characteristic polynomials in the circular Jacobi ensemble and Painlevé equations  (2608.24423 - Bothner et al., 25 Aug 2026) in Section 1, subsection “Painlevé equation for the characteristic function of \(\mathsf{X}_1(w)\)”, paragraph following Corollary \ref{almostsurely}