# 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.).

### Publications

*2D anisotropic KPZ at stationarity: Scaling, tightness and nontriviality*in The Annals of Probability

Description | Stochastic growth phenomena naturally emerge in a variety of physical and biological contexts, such as growth of combustion fronts or bacterial colonies, crystal growth on thin films by molecular beam epitaxy, turbulent liquid crystals, etc. The goal of researchers is to thoroughly understand the time evolution of these seemingly unrelated random surfaces so to be able to obtain accurate predictions on their behaviour. In one spatial dimension, over the last 40 years, the Mathematics and Physics communities in a joint effort determined what was widely believed to be the law of the only two universal processes presumed to capture the large-scale behaviour of such systems. In the preprint "The Brownian Castle" together with M. Hairer, we established the existence of a third universal process, the Brownian Castle. We provided a thorough construction of this process and determined its properties. In particular, we proved that the Brownian Castle is intimately connected to the Brownian Web, a family of coalescing Brownian motions starting from every space-time point in the plane, for which we provided a novel characterisation in the work "The Brownian Web as a random R-tree". The second major contribution of this research programme concerns the "critical" case of two-dimensional random surfaces. A particularly interesting class of models in this context is the so-called Anisotropic KPZ class whose elements exhibit three basic mechanisms: surface relaxation, local randomness and a dependence on the slope in which different directions have competing (and annihilating) effects. Together with D. Erhard and F. Toninelli, we focused on the paradigmatic model in the class, the AKPZ equation which is a singular stochastic partial differential equation. In the preprint "The stationary AKPZ equation: logarithmic superdiffusivity", we obtained a rather unexpected result, namely that the solution of the AKPZ equation is logarithmically superdiffusive, meaning that its correlation length diverges logarithmically at large scales. |

Exploitation Route | I have been presenting the outcomes of my work at conferences, dedicated workshops and seminars. The work "2D Anisotropic KPZ at stationarity: scaling, tightness and non-triviality" has been published in the leading journal of Probability, namely "Annals of Probability". The most recent works can be found on arXiv and on my personal website, so that all researchers can easily access them. |

Sectors | Other |