  A crucial role in the theory of uncertainty quantification (UQ) of PDEs is
played by the regularity of the solution with respect to the stochastic
parameters; indeed, a key property one seeks to establish is that the solution
is holomorphic with respect to (the complex extensions of) the parameters. In
the context of UQ for the high-frequency Helmholtz equation, a natural question
is therefore: how does this parametric holomorphy depend on the wavenumber $k$?
  The recent paper [Ganesh, Kuo, Sloan 2021] showed for a particular
nontrapping variable-coefficient Helmholtz problem with affine dependence of
the coefficients on the stochastic parameters that the solution operator can be
analytically 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 problems
with arbitrary (holomorphic) dependence on the stochastic parameters; we show
that in all cases the region of parametric holomorphy decreases with $k$, and
show how the rate of decrease with $k$ is dictated by whether the unperturbed
Helmholtz problem is trapping or nontrapping. We then give examples of both
trapping and nontrapping problems where these bounds on the rate of decrease
with $k$ of the region of parametric holomorphy are sharp, with the trapping
examples coming from the recent results of [Galkowski, Marchand, Spence 2021].
  An immediate implication of these results is that the $k$-dependent
restrictions imposed on the randomness in the analysis of quasi-Monte Carlo
(QMC) methods in [Ganesh, Kuo, Sloan 2021] arise from a genuine feature of the
Helmholtz equation with $k$ large (and not, for example, a suboptimal bound).
