Power variations of critical Gaussian multiplicative chaos and their spectral applications

Abstract: We introduce power variations of critical Gaussian multiplicative chaos (GMC) along refining partitions in arbitrary dimension. Under suitable renormalisation, we prove uniform fractional moment bounds via Laplace transform estimates as well as stable convergence to supercritical GMCs. Together, these results provide a unified approach to regularity and spectral questions for critical chaos, yielding: (i) the sharp exponent for the logarithmic modulus of continuity, (ii) the existence of non-empty essential spectrum for critical Liouville quantum gravity surfaces; and (iii) an almost-sure quantitative Fourier decay.