We consider frequency-domain acoustic scattering at a homogeneous star-shaped penetrable
object, whose shape is uncertain and modeled via a radial spectral parameterization with
random coefficients. We exploit 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] to obtain frequency-explicit statements on the holomorphic
dependence of the scattered field and the far-field pattern on the stochastic
parameters. This paves the way for applying general results on the efficient
construction of high-dimensional surrogate models. In addition, frequency-explicit
spatial regularity estimates permit us to quantify the impact of Galerkin
discretization using low-order Lagrangian finite-element spaces.