Large deviation theory

Large deviation theory studies the chances of rare events that significantly differ from typical behavior in complex systems. For example, it quantifies how unlikely it is for average outcomes—like the average temperature over a year—to deviate substantially from their usual levels. By analyzing these probabilities, the theory helps us understand and predict the likelihood of extreme events, such as financial crashes or natural disasters, even when they occur infrequently. It provides mathematical tools to assess risks and behaviors in fields like physics, finance, and engineering, emphasizing how unlikely deviations can still be systematically studied and understood.