2025 Theses Doctoral

Hydrodynamic limits and fluctuations for interacting particle systems

Drillick, Hindy

In this thesis, we study the limiting behavior of several different models in the KPZ universality class.

In Chapters 1 and 2, we study the hydrodynamic limits of the 𝑡-PNG model and the inhomogeneous stochastic six-vertex model with periodic weights using soft techniques. We show that the height functions of both models converge almost surely to a deterministic limit shape. The key element of the proofs is the construction of colored versions of both models, which allows us to apply the superadditive ergodic theorem. Along the way, we also construct the stationary 𝑡-PNG model and prove a version of Burke's theorem for it.

In Chapter 3, we consider the stochastic six-vertex model with step initial conditions and a single second-class particle at the origin. We show almost sure convergence of the speed of the second-class particle to a random limit. This allows us to define the stochastic six-vertex model speed process, whose law we show to be ergodic and stationary for the dynamics of the multi-class stochastic six-vertex process. The proof requires the development of precise bounds on the fluctuations of the height function of the stochastic six-vertex model around its limit shape using methods from integrable probability. As part of the proof, we also obtain a novel geometric stochastic domination result that states that a second-class particle to the right of any number of third-class particles will at any fixed time be overtaken by at most a geometric number of third-class particles.

In Chapters 4 and 5, we show that under a certain moderate deviation scaling, the multiplicative-noise stochastic heat equation describes the fluctuations of the quenched density of two different models of diffusion in a time-dependent random environment. The first model is the motion of a particle under a continuum random environment whose distribution is given by the Howitt-Warren flow. The second model is a 1D nearest-neighbor random walk in a space-time random environment. The proofs rely on certain Girsanov transforms and showing that the quenched density solves a prelimiting SPDE that resembles the stochastic heat equation. For the random walk model, we also show that independent noise is generated in the limit, in the sense that the prelimiting noise field does not converge to the driving noise of the limiting SPDE.