  A key quantity that occurs in the error analysis of several numerical methods
for eigenvalue problems is the distance between the eigenvalue of interest and
the next nearest eigenvalue. When we are interested in the smallest or
fundamental eigenvalue, we call this the spectral or fundamental gap. In a
recent manuscript [Gilbert et al., arXiv:1808.02639], the current authors,
together with Frances Kuo, studied an elliptic eigenvalue problem with
homogeneous Dirichlet boundary conditions, and with coefficients that depend on
an infinite number of uniformly distributed stochastic parameters. In this
setting, the eigenvalues, and in turn the eigenvalue gap, also depend on the
stochastic parameters. Hence, for a robust error analysis one needs to be able
to bound the gap over all possible realisations of the parameters, and because
the gap depends on infinitely-many random parameters, this is not trivial. This
short note presents, in a simplified setting, an important result that was
shown in the paper above. Namely, that, under certain decay assumptions on the
coefficient, the spectral gap of such a random elliptic eigenvalue problem can
be bounded away from 0, uniformly over the entire infinite-dimensional
parameter space.
