The combinatorics of KPZ at positive temperature

The Kardar-Parisi-Zhang equation (KPZ) was introduced in 1986 as a universal model to capture statistics of a wide range of physical phenomena such as growth of interfaces or turbulent fluids. Remarkably fluctuations of this class of systems fall out of the scope of the classical central limit theorem.

Understanding these phenomena has driven a tremendous activity in rigorous mathematics leading to groundbreaking theories or even to whole new fields such as that of Integrable Probability, where this proposal belongs.

Since the seminal work of Johansson (1999), it is understood that systems in the KPZ class are governed by distributions coming from random matrix theory. So far, a framework with clear ""determinantal structure"" has been created to tackle models at ""zero temperature"". Progress in positive temperature setting, including the KPZ equation, only came during the last decade.

Insights from many different fields (combinatorics, symmetric functions, etc) into probability, allowed to treat one-point statistics of certain systems at positive temperature.

In all instances a mysterious determinantal structure, whose origins elude understanding, appears to govern (so far only) one-point statistics.

This project aims to reveal the deep foundations of integrability of KPZ models at positive temperature and extend its scope.

This will allow to settle the solvability in situations that are currently out of reach such as restricted spatial geometries and multi-point correlations.

To achieve this we will follow a new route producing combinatorial mappings of positive temperature systems to purely determinantal ones.

Our approach will make novel uses of methods from combinatorics and integrable systems (via the Yang-Baxter toolbox) and will create new dynamics linking integrable systems (such as box-ball system) to the KPZ universe.

At the same time our probabilistic insights will give rise to new methodologies and will answer old questions from algebraic combinatorics