This proposal is on the interface of probability theory and rigorous aspects of statistical mechanics. The proposed activity aims at investigating the evolution of systems with complex interactions, such as particles moving in a disordered environment, cars navigating their way through traffic, the surface of a growing crystal, or the boundary of an infected tissue. Complexity is captured by the randomness in the model, both in the environment in which the particles interact or the crystal grows, and in the interaction or growth process itself. The aim of the project is to develop the mathematical laws that govern such systems. To have a very simple example in mind, one can think of how the fraction of Heads in a large number of tosses of a coin will converge to the probability of getting Heads in one toss. (The mathematical law behind this basic phenomenon is known as the Law of Large Numbers.) Besides its impact on probability theory and mathematics in general, the proposed activity is expected to have a direct impact on the understanding of many physical systems involving random motion in random or disordered media. Understanding complex interactions has wide implications for science and engineering and thereby for society.
The proposed activity is on the subject of random motion in random media. Models in this field have been intensely and concurrently studied by mathematicians, natural scientists, social scientists, and engineers. Their importance arises from rigorous mathematical connections to the celebrated Kardar-Parisi-Zhang (KPZ) equation, and the KPZ universality class. The PI has already established energy-entropy duality and produced solutions to these variational formulas in terms of Busemann functions. In particular, the PI proved almost sure existence of Busemann functions for the growth model with general weight distribution. The PI will next generalize this existence result to positive temperature polymers, as well as higher dimensional models. As a result, KPZ fluctuation exponents become at last accessible for general models, allowing to establish some universal properties for models beyond the narrow set of explicitly solvable cases. This also opens the door to establishing existence of Busemann functions in the continuum setting of the stochastic heat equation, enabling one to deduce global existence of a solution to the stochastic Burgers equation, as well as access KPZ fluctuation results directly rather than through approximation schemes via discrete models.
