Abstract
We explore statistical fluctuations over the ensemble of quantum trajectories in a model of two-dimensional free fermions subject to projective monitoring of local charge across the measurement-induced phase transition. Our observables are the particle-number covariance between spatially separated regions, G_AB, and the two-point density correlation function, C(r). Our results exhibit a remarkable analogy to Anderson localization, with G_AB corresponding to two-terminal conductance and C(r) to two-point conductance, albeit with different replica limit and unconventional symmetry class, geometry, and boundary conditions. In the delocalized phase, G_AB exhibits ``universal'', nearly Gaussian, fluctuations with variance of order unity. In the localized phase, we find a broad distribution of G_AB with -ln G_AB ∼ L (where L is the system size) and the variance var(ln G_AB) ∼ L^μ, and similarly for C(r), with μ ≈ 0.5. At the transition point, the distribution function of G_AB becomes scale-invariant and C(r) exhibits multifractal statistics, C^q(r)∼ r^-q(d+1) - Δ_q. We characterize the spectrum of multifractal dimensions Δ_q. Our findings lay the groundwork for mesoscopic theory of monitored systems, paving the way for various extensions.