Shannon entropy of the measurement record at measurement-dominated criticality and RG flow: A c-theorem for effective central charge and a g-theorem for effective boundary entropy

We present two theorems demonstrating non-perturbatively the decrease under relevant renormalization group (RG) flow of two quantities, c_{eff} and g_{eff} characterizing, respectively, the universal information content of the Shannon entropy of the measurement record for two different types of measurement-dominated criticality. We demonstrate the decrease of the effective central charge c_{eff} of 2D replica field theories in the R->1 replica limit that govern the long-distance physics of weakly monitored 2D classical critical systems (Baysian inference problems) studied recently in the literature [arXiv:2504.01264; arXiv:2504.12385; arXiv:2504.08888]. In particular, we show that c_{eff} is less than the central charge of the unmeasured critical system. We refer to this result as the c-effective theorem. In addition, we present an analogous g-effective theorem demonstrating the decrease under RG flow of the effective Affleck-Ludwig boundary entropy g_{eff}, quantifying a corresponding contribution to the Shannon entropy for analogous 2D defect replica field theories in the R->1 replica limit, which govern the long-distance physics of weak quantum measurements performed on one-dimensional quantum critical ground states.