Identifying symmetries in the statistical ensemble of coarse-graining rules

In statistical physical systems, symmetries which are absent at the level of microscopic building blocks often emerge in the collective behaviour. Moreover, without prior knowledge even microscopic symmetries may be difficult to identify from unstructured Monte Carlo or experimental data. Here we take an information theoretic perspective to address this challenge systematically. To this end we focus on real-space mutual information (RSMI), leveraging the crucial observation that coarse-graining transformations maximising RSMI can be formally identified as the relevant operators of the effective field theory. Using the recently introduced RSMI-NE algorithm, statistical ensembles of such coarse-graining transformations can be efficiently generated. In this work we study the information contained in this ensemble, and show how symmetries, broken and also emergent, can be identified. We also demonstrate the extraction of the phase diagram and the order parameters for equilibrium systems. Our approach paves the way towards automated data-driven discovery of emergent symmetries of complex statistical systems.