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A crucial role in the theory of uncertainty quantification (UQ) of PDEs isplayed by the regularity of the solution with respect to the stochasticparameters; indeed, a key property one seeks to establish is that the solutionis holomorphic with respect to (the complex extensions of) the parameters. Inthe context of UQ for the high-frequency Helmholtz equation, a natural questionis therefore: how does this parametric holomorphy depend on the wavenumber $k$?  The recent paper [Ganesh, Kuo, Sloan 2021] showed for a particularnontrapping variable-coefficient Helmholtz problem with affine dependence ofthe coefficients on the stochastic parameters that the solution operator can beanalytically continued a distance $\sim k^{-1}$ into the complex plane.  In this paper, we generalise the result in [Ganesh, Kuo, Sloan 2021] about$k$-explicit parametric holomorphy to a much wider class of Helmholtz problemswith arbitrary (holomorphic) dependence on the stochastic parameters; we showthat in all cases the region of parametric holomorphy decreases with $k$, andshow how the rate of decrease with $k$ is dictated by whether the unperturbedHelmholtz problem is trapping or nontrapping. We then give examples of bothtrapping and nontrapping problems where these bounds on the rate of decreasewith $k$ of the region of parametric holomorphy are sharp, with the trappingexamples coming from the recent results of [Galkowski, Marchand, Spence 2021].  An immediate implication of these results is that the $k$-dependentrestrictions imposed on the randomness in the analysis of quasi-Monte Carlo(QMC) methods in [Ganesh, Kuo, Sloan 2021] arise from a genuine feature of theHelmholtz equation with $k$ large (and not, for example, a suboptimal bound).

Wavenumber-explicit parametric holomorphy of Helmholtz solutions in the context of uncertainty quantification