Phase Transitions in a 2D Ising Model of Agent Expectations in Financial Markets: Analytics in One- and Two-Dimensional Network Topologies

Phase transitions between ordered and disordered states of interactive agents have been recognized as integral to dynamics in a range of economic and social processes. Several theorists in the study of financial markets have directly linked phase transitions between disordered and ordered states of agents to a critical point in the dynamics of market price. To date, phase transitions in the dynamics of price in financial markets have been demonstrated with numerical methods. In an application to a financial market, we propose a multicomponent in which a first component is in bounded rationality and a second component is in behavior that generates herding in financial markets. A transition function defines the relative weight of components. We extend conditions of Onsager (1944) for phase transitions in a 2D Ising model and analytically demonstrate that the proposed model evidences phase transitions. Generalizations of the results to other multi-component models are noted.