# The emergence of universal behaviour for growth models, stochastic PDEs and random operators.

Lead Research Organisation:
University of Warwick

Department Name: Statistics

### Abstract

In 1986, three physicists, Kardar, Parisi and Zhang, conjectured that all randomly evolving surfaces possessing three features, a smoothing mechanism, an underlying locally uncorrelated noise and a growth mechanism depending on the size of the slope, should have the same large-scale fluctuations, irrespective of their microscopic details. In other words, they predicted the existence of a Universality Class, that since then bares their name, and of a universal stochastic process, able to capture the behaviour of a wide class of models, such as turbulent liquid crystals, crystal growth on thin films, bacteria colony growth, etc. Over the last thirty years, their work stimulated the interest of a wide number of researchers, driven by the ambition to fully understand the nature of the KPZ Universality Class and to characterise this universal object. On the other hand, the Physics literature also predicts that, when a physical system possesses the same features apart from the slope dependence, then it belongs to a different Universality Class, the so-called Edwards-Wilkinson (EW) Universality Class, named after the two physicists that introduced it, and the universal process describing their behaviour is Gaussian and can be easily explicitly characterised.

The first objective of this research proposal is to show that in the context of (1+1)-dimensional (one for time and one for space) randomly evolving interfaces, the classification given above is not exhaustive and another Universality Class needs to be considered. Our goal is to rigorously construct the universal object at its core, a stochastic process called Growing Brownian Castle, determine its characterising properties, give the first instances of its universality and analyse its relation with KPZ.

In the context of the KPZ Universality Class, there is a model that plays a distinguished role and it is presumed to be universal itself. This model is a Stochastic Partial Differential Equation (SPDE), the KPZ Equation. Despite its importance, a satisfactory solution theory for this equation in one spatial dimension was established only recently thanks to the theory of Regularity Structures, by M. Hairer. The techniques that are now available allow for a systematic study of its universality and this research program intends to establish it for a family of models driven by conservative dynamic, which has never been considered so far.

For evolving surfaces in (1+2)-dimensions, the Universality Classes picture is subtler because the slope can evolve in different directions that could compete with each other. This proposal focuses on the case in which the contribution of the slope sizes in the different directions averages out. This class of models is called Anisotropic KPZ Universality Class and the long-standing conjecture, coming from the Physics literature, is that this class is nothing but EW in dimension 2. In other words it is expected that the slope does not play any role at all. The project aims at showing such a result for the Anisotropic KPZ Equation, a singular SPDE that cannot be treated by the theory of Regularity Structures mentioned above and for which radically new ideas are needed.

At last, the random operator we will focus on is the Anderson-Hamiltonian. Its importance lies on the fact that it is connected with the parabolic Anderson model, the scaling limit of random motion in random potential or branching processes in random media, and many others. We will determine some of its properties that will shed some light on its universal nature.

The first objective of this research proposal is to show that in the context of (1+1)-dimensional (one for time and one for space) randomly evolving interfaces, the classification given above is not exhaustive and another Universality Class needs to be considered. Our goal is to rigorously construct the universal object at its core, a stochastic process called Growing Brownian Castle, determine its characterising properties, give the first instances of its universality and analyse its relation with KPZ.

In the context of the KPZ Universality Class, there is a model that plays a distinguished role and it is presumed to be universal itself. This model is a Stochastic Partial Differential Equation (SPDE), the KPZ Equation. Despite its importance, a satisfactory solution theory for this equation in one spatial dimension was established only recently thanks to the theory of Regularity Structures, by M. Hairer. The techniques that are now available allow for a systematic study of its universality and this research program intends to establish it for a family of models driven by conservative dynamic, which has never been considered so far.

For evolving surfaces in (1+2)-dimensions, the Universality Classes picture is subtler because the slope can evolve in different directions that could compete with each other. This proposal focuses on the case in which the contribution of the slope sizes in the different directions averages out. This class of models is called Anisotropic KPZ Universality Class and the long-standing conjecture, coming from the Physics literature, is that this class is nothing but EW in dimension 2. In other words it is expected that the slope does not play any role at all. The project aims at showing such a result for the Anisotropic KPZ Equation, a singular SPDE that cannot be treated by the theory of Regularity Structures mentioned above and for which radically new ideas are needed.

At last, the random operator we will focus on is the Anderson-Hamiltonian. Its importance lies on the fact that it is connected with the parabolic Anderson model, the scaling limit of random motion in random potential or branching processes in random media, and many others. We will determine some of its properties that will shed some light on its universal nature.

### Planned Impact

Focusing on Probability Theory, Statistical Mechanics and Stochastic Analysis, this proposal will mainly benefit researchers in these areas since it answers many fundamental questions in the fields and develops a whole new spectrum of tools and techniques.

Although this research programme is on foundational Mathematics, it has the potential of influencing more applied sciences. For example, the Brownian Web, which is at the base of the new universal stochastic process I aim at introducing, and related constructions have already proved useful in population genetics. They allow to describe the complete space-time genealogy of a large population and to determine the spatial structure on the spread of a selectively advantageous gene through a population. The theory of Rough Paths, which represents the source out of which the new techniques in singular Stochastic PDEs (e.g. the Theory of Regularity Structures) were developed, is finding interesting and unexpected applications in Machine Learning and in particular in the recognition of characters. The precise formulas, obtained within the Mathematics community, concerning the exact fluctuations of models within the KPZ Universality Class motivated experimentalists to study those of liquid crystals.

In order to maximise the impact of my results, disseminate them among the widest possible audiences and initiate pathways leading to applications of my results outside of the Mathematics perimeter, I will do the following

1. Publish my results in high quality Mathematics journals (e.g. Annals of Probability, Communications in Mathematical Physics, Communications in Pure and Applied Mathematics, Probability Theory and Related Fields, etc.), write expository and non-specialist articles,

2. Present my work at relevant international conferences (Conference on Stochastic Processes and Applications, EquaDiff, International Congress on Mathematical Physics, etc.), specialised workshops and seminars, as well as give talks at other departments and at the university open days (for example, the Imperial Festival),

3. Organise a week-long international workshop at Imperial College London, involving the most prominent mathematicians working on the fields this proposal centres on as well scientists from other disciplines (Biology and Physics),

4. Collaborate with researchers from other institutions, Khalil Chouk (Technische Universität Berlin), Dirk Erhard (Universidade Federal de Bahia, Brazil), Nicolas Perkowski (Humboldt-Universität zu Berlin) and Nikolaos Zygouras (University of Warwick),

5. Coordinate the Stochastic Analysis Reading Seminar at Imperial College London and teach an EPSRC-Taught Course Centre course on topics related to this proposal (e.g. Brownian Web and its properties, Singular Stochastic PDEs, the Parabolic Anderson Model, etc.).

Although this research programme is on foundational Mathematics, it has the potential of influencing more applied sciences. For example, the Brownian Web, which is at the base of the new universal stochastic process I aim at introducing, and related constructions have already proved useful in population genetics. They allow to describe the complete space-time genealogy of a large population and to determine the spatial structure on the spread of a selectively advantageous gene through a population. The theory of Rough Paths, which represents the source out of which the new techniques in singular Stochastic PDEs (e.g. the Theory of Regularity Structures) were developed, is finding interesting and unexpected applications in Machine Learning and in particular in the recognition of characters. The precise formulas, obtained within the Mathematics community, concerning the exact fluctuations of models within the KPZ Universality Class motivated experimentalists to study those of liquid crystals.

In order to maximise the impact of my results, disseminate them among the widest possible audiences and initiate pathways leading to applications of my results outside of the Mathematics perimeter, I will do the following

1. Publish my results in high quality Mathematics journals (e.g. Annals of Probability, Communications in Mathematical Physics, Communications in Pure and Applied Mathematics, Probability Theory and Related Fields, etc.), write expository and non-specialist articles,

2. Present my work at relevant international conferences (Conference on Stochastic Processes and Applications, EquaDiff, International Congress on Mathematical Physics, etc.), specialised workshops and seminars, as well as give talks at other departments and at the university open days (for example, the Imperial Festival),

3. Organise a week-long international workshop at Imperial College London, involving the most prominent mathematicians working on the fields this proposal centres on as well scientists from other disciplines (Biology and Physics),

4. Collaborate with researchers from other institutions, Khalil Chouk (Technische Universität Berlin), Dirk Erhard (Universidade Federal de Bahia, Brazil), Nicolas Perkowski (Humboldt-Universität zu Berlin) and Nikolaos Zygouras (University of Warwick),

5. Coordinate the Stochastic Analysis Reading Seminar at Imperial College London and teach an EPSRC-Taught Course Centre course on topics related to this proposal (e.g. Brownian Web and its properties, Singular Stochastic PDEs, the Parabolic Anderson Model, etc.).

Description | In the preprint "2D Anisotropic KPZ at stationarity: scaling, tightness and non-triviality" with D. Erhard and P. Schönbauer, we derive the first available rigorous results on the 2D anisotropic KPZ equation. This is a stochastic partial differential equation which is conjectured to describe the universal fluctuations of a large class of two dimensional random surfaces. Despite its importance from both a physical and mathematical viewpoint, nothing rigorous had been shown. This is due to the fact that the equation is extremely singular and even the most powerful (and recent) techniques developed in the work of the fields medallist M. Hairer fail to provide a notion of solution for it. In the present work, we identify a (family) of scaling regimes for which a suitable approximation can be shown admit subsequential limits and determine the first properties of these limit points. |

Exploitation Route | The paper has been accepted in the leading journal of Probability, namely "Annals of Probability", so that the community has been made aware of the results. Moreover, I have been and will continue to present my work at relevant conferences, workshops and seminars. |

Sectors | Other |