  In this paper we establish a mathematically rigorous connection between
Causal inference (C-inf) and the low-rank recovery (LRR). Using Random Duality
Theory (RDT) concepts developed in [46,48,50] and novel mathematical strategies
related to free probability theory, we obtain the exact explicit typical (and
achievable) worst case phase transitions (PT). These PT precisely separate
scenarios where causal inference via LRR is possible from those where it is
not. We supplement our mathematical analysis with numerical experiments that
confirm the theoretical predictions of PT phenomena, and further show that the
two closely match for fairly small sample sizes. We obtain simple closed form
representations for the resulting PTs, which highlight direct relations between
the low rankness of the target C-inf matrix and the time of the treatment.
Hence, our results can be used to determine the range of C-inf's typical
applicability.
