KPZcomb · The combinatorics of KPZ at positive temperature
Horizon 2020 — Marie Skłodowska-Curie Actions
- Duration
- 2021-09-02 → 2023-09-01
- EU contribution
- €224,934
- Participants
- 1
- Scheme
- MSCA-IF
Lines connect the coordinator with its partners.
Results in brief
The combinatorics of KPZ at positive temperature
This project aimed at building up a combinatorial theory of stochastic solvable growth models of one-dimensional interfaces. The aim of the proposed theory is to deepen the understanding of these universal processes and producing for them exact and rigorous mathematical descriptions. Practical examples are mathematical models of growth, such as fire or bacteria spreading. In the last decade several breakthroughs have allowed a deep mathematical understanding of these phenomena. In this project we have focused on several outstanding problems that had remained open. These include the study of growth in presence of boundaries, which in practical applications one can think of areas where fire of bacteria cannot spread, and the study of rare events beyond typical fluctuations. Throughout the development of the project several of these long standing open problems were tackled and solved. In parallel new original results were developed leading to new questions and promising research directions.
Data: CORDIS, © European Union
Project objective
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""
Original text from CORDIS.
Participants
- UNIVERSITY OF WARWICK · COVENTRYCoordinatorUnited Kingdom
Links
Data: CORDIS, © European Union