Microscopic Origin of Hyperuniformity in Dense Random-Organizing Systems

Random-organizing systems---non-equilibrium systems displaying a phase transition between an absorbing and an active phase---have been extensively studied for their fundamental importance and practical relevance, with applications spanning driven colloidal suspensions, random close packing, and two-dimensional crystallization. Although the critical behavior of such systems has been well characterized, a microscopic understanding of the active phase remains elusive, in particular, it is unclear how short-range microscopic dynamics lead to long-range macroscopic structure. Here, combining simulations, and fluctuating hydrodynamic theory, we study the active phase of random-organizing systems. We demonstrate how short-range noise correlations govern long-range density fluctuations and reveal a scale-dependent hyperuniformity in random organizing systems, where the system is strongly hyperuniform below a critical length scale, and is random above it. Our results quantitatively explain the active phase behavior of all random-organizing systems, namely random organization (RandOrg), biased random organization (BRO), and additionally, systems which can be mapped to random-organizing systems, such as stochastic gradient descent (SGD). Overall, our study highlights the crucial role of noisy interactions in a generic class of out-of-equilibrium systems and is simultaneously relevant for robust, and controllable self-assembly of hyperuniform structures.