We study the cosmology of axions in the post-inflationary scenario, where random initial conditions and the ensuing string-domain-wall network generate an isocurvature power spectrum. For subdominant axion fractions, the resulting white-noise fluctuations enhance the growth of structure in the dominant component. Combining ultraviolet luminosity functions of galaxies with Lyman-$\alpha$ and CMB data yields leading cosmological limits on post-inflationary axion dark matter, linking early-universe field dynamics to high-redshift structure formation.