Large Deviations: Theory and Applications

This lecture explores the concept of large deviations in statistical mechanics, connecting it to empirical means of random variables and the moment generating function. Starting with the basics of random variables, the instructor delves into Kramer's theorem, which serves as the foundation of large deviation theory. The discussion extends to cardinal arrays and Varadhan's lemma, showcasing how to estimate probabilities and expectations of exponential functions using the rate function. The lecture also touches upon Laplace's method and its application in estimating expectations. By the end, the audience gains insights into the rigorous version of Laplace's method and its broader applicability beyond exponential functions, providing a comprehensive understanding of large deviation theory.