Universality Classes for the KPZ Fixed Point and Large Deviation Principles for Interacting Particle Systems

Mathematical Applications of Science Fiction We establish a rigorous derivation of the Large DeviationPrinciple (LDP) for the Kardar-Parisi-Zhang (KPZ) fixedpoint, viewed as the universal scaling limit of the TotallyAsymmetric Simple Exclusion Process (TASEP) under the1:2:3 scaling. By employing a transfinite renormalizationprocedure on the associated Fredholm determinants and uti-lizing the integrable structure of the Airy line ensemble, weconstruct the rate function I explicitly in terms of a varia-tional problem involving the polylogarithm of the scatteringdata. We prove that the limiting height function hKPZ(t, x)satisfies a Freidlin-Wentzell type LDP with speed ϵ−1 inthe space of upper semi-continuous functions endowed withthe topology of hypoelliptic convergence. Furthermore, wedemonstrate the existence of a unique minimizer for thevariational action, thereby resolving the uniqueness of themacroscopic fluctuation path. Our approach unifies the alge-braic Bethe ansatz techniques with the analytic frameworkof regularity structures.