  We study the dynamics of a spectator Higgs field which stochastically evolves
during inflation onto near-critical trajectories on the edge of a runaway
instability. We show that its fluctuations do not produce primordial black
holes (PBHs) in sufficient abundance to be the dark matter, nor do they produce
significant second-order gravitational waves. First we show that the Higgs
produces larger fluctuations on CMB scales than on PBH scales, itself a no-go
for a viable PBH scenario. Then we track the superhorizon perturbations
nonlinearly through reheating using the delta N formalism to show that they are
not converted to large curvature fluctuations. Our conclusions hold regardless
of any fine-tuning of the Higgs field for both the Standard Model Higgs and for
Higgs potentials modified to prevent unbounded runaway.
