Topological phases in discrete stochastic systems

Topological invariants have proved useful for analyzing emergent function as they characterize a property of the entire system, and are insensitive to local details, disorder, and noise. They support boundary states, which reduce the system response to a lower dimensional space and, in two-dimensional (2D) systems, offer a mechanism for the emergence of global cycles within a large phase space. Topological invariants have been heavily studied in quantum electronic systems and have been observed in other classical platforms such as mechanical lattices. However, this framework largely describes equilibrium systems within an ordered crystalline lattice, whereas biological systems are often strongly non-equilibrium with stochastic components. We review recent developments in topological states in discrete stochastic models in one-dimensional and 2D systems, and initial progress in identifying testable signatures of topological states in molecular systems and ecology. These models further provide simple principles for targeted dynamics in synthetic systems and in the engineering of reconfigurable materials. Lastly, we describe novel theoretical properties of these systems such as the necessity for non-Hermiticity in permitting edge states, as well as new analytical tools to reveal these properties. The emerging developments shed light on fundamental principles for non-equilibrium systems and topological protection enabling robust biological function.