IS31: Random Growth and KPZ Universality

Organizer: Shirshendu Ganguly (UC Berkeley)

Two-layer Gibbs line ensembles

Guillaume Barraquand

Stationary measures of a variety of growth models in the Kardar-Parisi-Zhang class can be described in terms of couples of interacting random walks or Brownian motions, called two-layer Gibbs measures. The Boltzmann weights used to define them originate in branching rules satisfied by families of symmetric functions in the Macdonald hierarchy (Schur polynomials, Hall-Littlewood functions, (q)-Whittaker...). This framework applies to various models: interacting particle systems between boundary reservoirs, directed polymer models in a strip, last passage percolation, the stochastic six-vertex model. This talk will present an overview of the method, as well as more recent results on matrix valued growth processes.

On the global solutions of the KPZ fixed point

Ofer Busani

The Kardar–Parisi–Zhang (KPZ) fixed point is a fundamental random process that arises as the universal scaling limit for a broad class of 1+1-dimensional stochastic growth models. A key feature of the KPZ fixed point is the conservation of the slope of the initial condition, which prompts the natural question: for a fixed realization of the system, how many global (bi-infinite in time) solutions exist with a given slope?

Previous work \([2,1]\) has established the existence of a “good” set of slopes for which the “one force–one solution” principle applies—namely, each such slope yields a unique global solution. However, beyond this good set, little is known. In particular, while it is known that uniqueness fails for “bad” slopes, the number and structure of these non-unique global solutions is unknown.

In this work, we provide a complete characterization of global solutions to the KPZ fixed point with a prescribed slope. In particular, we show that for every bad slope, there exist infinitely many distinct global solutions. At the heart of the proof is a bijection between global solutions and certain bi-infinite interfaces.

Joint work with Sudeshna Bhattacharjee and Evan Sorensen

Bibliography

\([1]\) Busani, Ofer and Seppäläinen, Timo and Sorensen, Evan (2024). The stationary horizon and semi-infinite geodesics in the directed landscape. The Annals of Probability.

\([2]\) Rahman, Mustazee and Virág, Bálint (2025). Infinite geodesics, competition interfaces and the second class particle in the scaling limit. Annales de l’Institut Henri Poincare (B) Probabilites et statistiques.

Scaling limit of half-space KPZ equation

Sayan Das

We consider the KPZ equation in the half-space: \[\partial_t\mathcal{H} = \frac12\partial_x^2\mathcal{H}+\frac12(\partial_x \mathcal{H})^2+\xi, \quad (x,t)\in \mathbb{R}_{\ge0}^2\] with narrow wedge initial data and Neumann boundary condition: \[\partial_x\mathcal{H}(x,t)\mid_{x=0}=\alpha\] In the talk, I will discuss the process level scaling limit of \(\mathcal{H}(x,t)\) as \(t\to \infty\) under the 1:2:3 KPZ scaling. This is based on a joint work with Christian Serio.

KPZ equation from some interacting particle systems

Kevin Yang

The focus of this talk is the derivation of the KPZ equation as a continuum limit for a large class of interacting particle systems given by exclusion processes with speed-changed drifts. A common feature of these particle systems is a lack of explicit invariant measures in general. This is also the source of the key technical difficulties that are ultimately resolved by analysis of the associated Kolmogorov equations. Time permitting, we will discuss applications of the methods to related settings, such as the open KPZ equation and open ASEP.