The weak noise theory (WNT) provides a framework for accessing large deviations in models of the Kardar-Parisi-Zhang (KPZ) universality class, probing the regime where randomness is small, fluctuations are rare, and atypical events dominate. Historically, two methods have been available: asymptotic analysis of Fredholm determinant formulas—applicable only for special initial data—and variational or saddle-point formulations leading to nonlinear evolution equations, which were mostly accessible perturbatively.
This talk explains how these approaches can be unified: the weak-noise saddle equations of KPZ-class models form classically integrable systems, admitting Lax pairs, conserved quantities, and an inverse scattering framework. In this setting, the large-deviation rate functions arise directly from the conserved charges of the associated integrable dynamics.
The discussion will focus on three examples:
1. The scalar Strict-Weak polymer ;
2. A matrix Strict-Weak polymer driven by Wishart noise ;
3. If time permits, the continuous-time q-TASEP.