Extreme Synergy in the Random-Energy Model

The random-energy model (REM), a solvable spin-glass model, has impacted an incredibly diverse set of problems, from protein folding to combinatorial optimization, to many-body localization. Here, we explore a new connection to secret sharing. We derive an analytic expression for the mutual information between any two disjoint thermodynamic subsystems of the REM. Our analyses reveal that the correlations in the REM exhibit extreme synergy, equivalent to that in a secure secret-sharing scheme. We formulate a secret-sharing scheme based on the REM and determine the ranges of temperatures and secret lengths over which the REM satisfies the requirement of secure secret sharing. We show further that a special point in the phase diagram exists at which the REM-based scheme is physically optimal in its information encoding. Our results for the thermodynamic limit are in good qualitative agreement with numerical simulations of finite systems, for which the strict security requirement is replaced by a tradeoff between secrecy and recoverability. Our work offers a new language to characterize synergistic correlations in many-body systems and a further example of information theory as a unifying concept, connecting problems in statistical physics to those in computation.