The combinatorics of KPZ at positive temperature

Project Information

KPZcomb

Grant agreement ID: 101030938

DOI

10.3030/101030938

Project closed

EC signature date 29 April 2021

Start date 2 September 2021

End date 1 September 2023

Funded under

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Total cost

€ 224 933,76

EU contribution

€ 224 933,76

Coordinated by

UNIVERSITY OF WARWICK
United Kingdom

Periodic Reporting for period 1 - KPZcomb (The combinatorics of KPZ at positive temperature)

Reporting period: 2021-09-02 to 2023-09-01

Summary of the context and overall objectives of the project

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.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

The proposal comprised three main scientific objectives, all of which have been fully achieved.

The first scientific objective consisted in producing a precise relationship between growth models of interfaces and determinantal point processes, a class of probabilistic systems with a simpler and clearer mathematical structure. Investigation towards the realization of this scientific objective are carried out in two preprints (arXiv:2106.11913 arXiv:2307.01179) which are currently under review for scientific publication.

The second scientific objective consisted in developing a combinatorial framework to explain correspondences discovered in the first scientific objective. This led to the development of vast generalizations of celebrated and classical results in algebraic combinatorics such as the Robinson-Schensted-Knuth correspondence. Results produced toward the realization of this second scientific objective are contained in the scientific paper Forum of Mathematics, Pi 11, e27 and in the preprint arXiv:2301.00132 which is currently under review for scientific publication.

The third scientific objective, consisted in applying the combinatorial methods developed to the study of growing interfaces. Precise mathematical formulas descending from the new combinatorial framework have led to a rigorous description of the celebrated Kardar-Parisi-Zhang stochastic differential equation including the case when the growing interface interacts with a boundary. Results produced towards the completion of this third scientific objective are contained in the preprints arXiv:2204.08420 arXiv:2307.01179 which are currently under review and in the scientific publication J. Math. Phys. 64, 083301.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

Results stemming from this project have been impactful mostly on the scientific community as important breakthrough towards the understanding of mathematical and physical phenomena around randomness in growth processes.

The characterization of growth processes with boundary interactions, performed in arXiv:2204.08420 has quickly become a standard reference in the field. Ideas generated within the context of this work have led to a plethora of applications and spurred the research activity in probability and stochastic analysis. Combinatorial ideas developed in the article Forum of Mathematics, Pi 11, e27, have been adapted and extended in several directions, both for their use in the study of interacting particle processes, but also in the context of representation theory. Furthermore, the techniques developed in arXiv:2307.01179 to study rare events in growth processes promise to become ameanable to extension and generalization in multple directions as they provide unprecedented connections between theories in probability and seemingly unrelated fields such as algebraic geometry.

A plot of the lower tail rate function of the deformed polynuclear growth.

Visualization of the deformed polynuclear growth.

An example of the skew RSK map

Random polymers in random environments with different geometries.