# Structures and universalities around the Kardar-Parisi-Zhang equation

Lead Research Organisation:
University of Warwick

### Abstract

It was proposed by Kardar, Parisi and Zhang in the 1980s that a large class of randomly growing interfaces exhibit universal fluctuations described mathematically by a nonlinear stochastic partial differential equation, which is now known as the Kardar-Parisi-Zhang or KPZ equation. Models within this class exhibit three basic mechanisms: growth as a function of the steepness of the interface, a smoothing effect modelled by Laplacian and local randomness modelled by white noise. Examples of physical systems which fall in this class are percolation of liquid in porous media, growth of bacteria colonies, currents in one dimensional traffic or liquid systems, liquid crystals etc.

Remarkably the fluctuations of such random interfaces are governed by exponents and distributions that differ from the predictions given by the classical central limit theorem. In dimension one they are, surprisingly, linked to laws emerging from random matrix theory, as this was first exhibited by the work of Baik-Deift-Johansson, followed by a flurry of activity which set the framework of "determinantal processes". New exciting developments have taken place in the more recent years, making the first important steps into universality beyond determinantal models. In dimension two the situation is much less developed as governing exponents and distributions are not known and even the meaning of the two dimensional KPZ is not set in place.

The goal of the project is twofold:

A. To penetrate deeper into the structure of one dimensional KPZ via setting a robust framework to study fluctuations of non determinantal systems, attacking pending conjectures on multipoint correlations and exploring new grounds into the universality and localisation phenomena. In doing so, novel links between probability, algebraic combinatorics, random matrix theory, integrable systems, number theory (automorphic forms) will be made.

B. To make the first steps in dimension two by constructing, via suitable scaling limits of discrete systems, the object(s) that incarnate the two dimensional KPZ equation and extract their properties.

Remarkably the fluctuations of such random interfaces are governed by exponents and distributions that differ from the predictions given by the classical central limit theorem. In dimension one they are, surprisingly, linked to laws emerging from random matrix theory, as this was first exhibited by the work of Baik-Deift-Johansson, followed by a flurry of activity which set the framework of "determinantal processes". New exciting developments have taken place in the more recent years, making the first important steps into universality beyond determinantal models. In dimension two the situation is much less developed as governing exponents and distributions are not known and even the meaning of the two dimensional KPZ is not set in place.

The goal of the project is twofold:

A. To penetrate deeper into the structure of one dimensional KPZ via setting a robust framework to study fluctuations of non determinantal systems, attacking pending conjectures on multipoint correlations and exploring new grounds into the universality and localisation phenomena. In doing so, novel links between probability, algebraic combinatorics, random matrix theory, integrable systems, number theory (automorphic forms) will be made.

B. To make the first steps in dimension two by constructing, via suitable scaling limits of discrete systems, the object(s) that incarnate the two dimensional KPZ equation and extract their properties.

### Planned Impact

Long term prosperity of every country depends crucially on a strong scientific base and in particular on foundational research. The high quality outcomes of this proposal will enhance and solidify these foundations thus promoting long term socioeconomic growth.

The proposal will raise the UK's scientific status in a highly regarded and competitive research area. The primary impact of the proposed work will be within probability as I plan to mix ideas from probability (stochastic analysis, random matrices, interacting particle systems, random walks) and outside (algebraic combinatorics, integrable systems, automorphic forms) to make crucial steps in the understanding of KPZ universality and its relations to a wide range of disciplines. But I also expect the impact to be felt outside probability as it is likely that the outcomes of my research will be of interest to these other disciplines.

In the long run, the outcomes of this proposal are likely to find their way into more applied disciplines, as is usually the case with high quality foundational research. Twenty-five years after the non-rigorous, theoretical physics work of Kardar-Parisi-Zhang (KPZ) on the fluctuations of randomly growing surfaces, experimentalists were motivated by the rigorous follow up works by mathematicians to search for and detect the precise type of fluctuations in liquid crystals. It is therefore possible that, in the long term, my work could be of interest to more applied scientists.

Towards these goals, the pathways to impact that I will take are:

1) Publish in high quality journals and participate in international conferences beyond the scope my core discipline.

2) Organise two workshops on the interface of probability with other fields.

3) Enhance my engagement with the Complexity Science Centre at Warwick and the EPSRC funded Doctoral Training Centre for Real World Systems, which provides opportunities to establish new contacts in other disciplines, for example through the seminar programme and joint supervision.

4) Nurture new talent via supervision of MSc/Phd theses, both through the doctoral training centres and through my overseas contacts.

Outreach: I also plan to engage in outreach activities. I have already contributed in this direction via speaking at high schools on topics of the current proposal, explaining the wide scope of probability and the notion of universality (including KPZ universality). Engagement with such programmes is planned to continue and expand.

Opportunities into providing scientific advice outside the academic field will be sought. A venue like this will be through my recent contacts with members of the British Science Museum.

The proposal will raise the UK's scientific status in a highly regarded and competitive research area. The primary impact of the proposed work will be within probability as I plan to mix ideas from probability (stochastic analysis, random matrices, interacting particle systems, random walks) and outside (algebraic combinatorics, integrable systems, automorphic forms) to make crucial steps in the understanding of KPZ universality and its relations to a wide range of disciplines. But I also expect the impact to be felt outside probability as it is likely that the outcomes of my research will be of interest to these other disciplines.

In the long run, the outcomes of this proposal are likely to find their way into more applied disciplines, as is usually the case with high quality foundational research. Twenty-five years after the non-rigorous, theoretical physics work of Kardar-Parisi-Zhang (KPZ) on the fluctuations of randomly growing surfaces, experimentalists were motivated by the rigorous follow up works by mathematicians to search for and detect the precise type of fluctuations in liquid crystals. It is therefore possible that, in the long term, my work could be of interest to more applied scientists.

Towards these goals, the pathways to impact that I will take are:

1) Publish in high quality journals and participate in international conferences beyond the scope my core discipline.

2) Organise two workshops on the interface of probability with other fields.

3) Enhance my engagement with the Complexity Science Centre at Warwick and the EPSRC funded Doctoral Training Centre for Real World Systems, which provides opportunities to establish new contacts in other disciplines, for example through the seminar programme and joint supervision.

4) Nurture new talent via supervision of MSc/Phd theses, both through the doctoral training centres and through my overseas contacts.

Outreach: I also plan to engage in outreach activities. I have already contributed in this direction via speaking at high schools on topics of the current proposal, explaining the wide scope of probability and the notion of universality (including KPZ universality). Engagement with such programmes is planned to continue and expand.

Opportunities into providing scientific advice outside the academic field will be sought. A venue like this will be through my recent contacts with members of the British Science Museum.

## People |
## ORCID iD |

Nikolaos Zygouras (Principal Investigator / Fellow) |

### Publications

Bisi E
(2022)

*Transition between characters of classical groups, decomposition of Gelfand-Tsetlin patterns and last passage percolation*in Advances in Mathematics
Bisi E
(2023)

*Non-intersecting Path Constructions for TASEP with Inhomogeneous Rates and the KPZ Fixed Point.*in Communications in mathematical physics
Bisi E
(2021)

*The geometric Burge correspondence and the partition function of polymer replicas*in Selecta Mathematica
Caravenna F
(2020)

*The two-dimensional KPZ equation in the entire subcritical regime*in The Annals of Probability
Caravenna F
(2019)

*The Dickman subordinator, renewal theorems, and disordered systems*in Electronic Journal of Probability
Caravenna F
(2019)

*On the Moments of the $$(2+1)$$-Dimensional Directed Polymer and Stochastic Heat Equation in the Critical Window*in Communications in Mathematical Physics
Lygkonis D
(2023)

*Moments of the 2D Directed Polymer in the Subcritical Regime and a Generalisation of the Erdös-Taylor Theorem*in Communications in Mathematical Physics
Lygkonis D
(2022)

*Edwards-Wilkinson fluctuations for the directed polymer in the full L2-regime for dimensions d=3*in Annales de l'Institut Henri PoincarÃ©, ProbabilitÃ©s et StatistiquesDescription | The project studies (i) stochastic integrability of models of stochastic growth such as colonies of bacteria, propagation of fluids in porous media etc, (ii) how disorder affects the properties of statistical mechanics models. The work within this project has led to confirmations of physical predictions and new points of view of physical theories thus creating a robust framework for further predictions and investigations. |

Exploitation Route | In mathematics: the methods have bridged disparate fields of mathematics, showing surprising links. In physics: the outcome have built a solid theory that facilitate further predictions. |

Sectors | Education |

Description | The findings have been communicated to high school students |

First Year Of Impact | 2019 |

Sector | Education |

Impact Types | Societal |