Abstract: The KPZ universality class is a broad collection of mathematical and physical models linked through their shared universal scaling behavior. A privileged subset, known as exactly solvable models, possess algebraic structure that allows for exact formulas for their probability distributions.

For an even smaller subset of these models, their distributions have been shown to satisfy nonlinear equations from classical integrable systems, including Painlevé, KP, and Toda equations. Why should equations from soliton theory appear in random growth, and why do they keep reappearing across different models?

I will describe a framework that gives a common algebraic explanation and extends this connection to eighteen exactly solvable models. The framework organizes the Fredholm determinant formulas of solvable KPZ models into an overdetermined linear problem whose compatibility conditions reveal a discrete integrable structure gauge equivalent to the non-abelian Hirota–Miwa equation.