  We consider frequency-domain acoustic scattering at a homogeneous star-shaped
penetrable obstacle, whose shape is uncertain and modelled via a radial
spectral parameterization with random coefficients. Using recent results on the
stability of Helmholtz transmission problems with piecewise constant
coefficients from [A. Moiola and E. A. Spence, Acoustic transmission problems:
wavenumber-explicit bounds and resonance-free regions, Mathematical Models and
Methods in Applied Sciences, 29 (2019), pp. 317-354] we obtain
frequency-explicit statements on the holomorphic dependence of the scattered
field and the far-field pattern on the stochastic shape parameters. This paves
the way for applying general results on the efficient construction of
high-dimensional surrogate models. We also take into account the effect of
domain truncation by means of perfectly matched layers (PML). In addition,
spatial regularity estimates which are explicit in terms of the wavenumber $k$
permit us to quantify the impact of finite-element Galerkin discretization
using high-order Lagrangian finite-element spaces.
