Dmitry Belyaev

Professor of Mathematics
Tutorial Fellow at St.Anne's College
e-mail: belyaev@maths.ox.ac.uk
phone: +44 1865 615154
Office: Andrew Wiles Building S1.20

My main research interests are on the interface between analysis and probability. In particular I am interested in:
• Geometry of Gaussian fields. Zeros of random functions, random plane waves.
• Growth models such as Diffusion Limited Aggregation and Hastings-Levitov models.
• Geometric function theory, boundary behaviour of conformal maps, coefficient problems.
• Fine structure of harmonic measure, multifractal formalism.
• Schramm-Loewner Evolution, critical lattice models.

Lectures

MT17: Lecturer for C4.8 Complex Analysis: conformal maps and geometry. Lecture notes and problem sheets are available at course page

Projects and Theses

I am supervising some BSP projects on topics related to my research. More information about BSP and possible topics could be found here

Please contact me by e-mail if you are interested in writing a Master Thesis or Part B or C dissertation under my supervision. I can suggest a number of topics in Analysis and Probability

Master students

• Mads Hansen Joint supervision with Prof. S. Waters. Thesis "Diffusive Transport in Complex Geometries"
• Meen-Sung Kim Thesis "A study of the long-time behaviour of stationary Diffusion Limited Aggregation"
• Zeljko Kereta Joint supervision with Prof. N. Trefethen. Thesis "Numerical study into validity of the Bogomolny-Schmit conjecture".

Preprints and presentations

• D. Beliaev, M. McAuley, S. Muirhead On the number of excursion sets of planar Gaussian fields Available at: arXiv The Nazarov-Sodin constant describes the average number of nodal set components of Gaussian fields on large scales. We generalise this to a functional describing the corresponding number of level set components for arbitrary levels. Using results from Morse theory, we express this functional as an integral over the level densities of different types of critical points, and as a result deduce the absolute continuity of the functional as the level varies. We further give upper and lower bounds showing that the functional is at least bimodal for certain isotropic fields, including the important special case of the random plane wave.
• D. Beliaev, S. Muirhead, I. Wigman Russo-Seymour-Welsh estimates for the Kostlan ensemble of random polynomials Available at: arXiv We study crossing probabilities for the nodal set of the Kostlan ensemble of random homogeneous polynomials on the sphere. Our main result is a Russo-Seymour-Welsh (RSW) type estimate, that is, a lower bound on the probability that the nodal set crosses a quad that is uniform in the degree of the polynomial, the conformal type' of the quad, and on all relevant scales. Our work extends recent results by Beffara-Gayet for the local scaling limit of this ensemble; the main differences are that (i) our result is valid before passing to the limit, (ii) we work directly on the sphere, and (iii) our methods are able to handle small negative correlations, incompatible with the techniques of Tassion as applied in Beffara-Gayet. In a more general setting, we establish RSW estimates for the nodal sets of sequences of Gaussian random fields defined on either the sphere or the flat torus under the following assumptions: (i) sufficient symmetry; (ii) smoothness and non-degeneracy; (iii) local convergence of the covariance kernels; (iv) asymptotically non-negative correlations; and (v) uniform rapid decay of correlations.
• Slides from my talk at UK Easter Probability meeting. In this talk I explained the conjectured relation between the nodal domains of a random plane wave and critical percolation.

Published and accepted papers

• M. Chavent, A.L. Duncan, P. Rassam, O. Birkholz, J. Hélie, T. Reddy, D. Beliaev, B. Hambly, J. Piehler, C. Kleanthous and M.S.P. Sansom How nanoscale protein interactions determine the mesoscale dynamic organisation of bacterial outer membrane proteins Nature Communication 2018 Jul 20;9(1):2846. Available at Nature: open access The spatiotemporal organisation of membranes is often characterised by the formation of large protein clusters. In Escherichia coli, outer membrane protein (OMP) clustering leads to OMP islands, the formation of which underpins OMP turnover and drives organisation across the cell envelope. Modelling how OMP islands form in order to understand their origin and outer membrane behaviour has been confounded by the inherent difficulties of simulating large numbers of OMPs over meaningful timescales. Here, we overcome these problems by training a mesoscale model incorporating thousands of OMPs on coarse-grained molecular dynamics simulations. We achieve simulations over timescales that allow direct comparison to experimental data of OMP behaviour. We show that specific interaction surfaces between OMPs are key to the formation of OMP clusters, that OMP clusters present a mesh of moving barriers that confine newly inserted proteins within islands, and that mesoscale simulations recapitulate the restricted diffusion characteristics of OMPs.
• A. Krause, D. Beliaev, R. Van Gorder, S. Waters Analysis of Lattice and Continuum Models of Bioactive Porous Media To appear in International Journal of Bifurcation and Chaos. Available at: arXiv In the context of tissue engineering, we recently proposed a lattice model for a bioactive porous tissue scaffold in order to understand the role of an active pore network in tissue growth [Lattice and Continuum Modelling of a Bioactive Porous Tissue Scaffold, preprint, 2017]. This model considered the scaffold as an evolving lattice of pores, with coupling between local cell growth in the pores, and fluid flow through the medium. Here we consider a variant of this lattice model as well as a spatially continuous analogue. We analyze these models from a dynamical systems perspective emphasizing qualitative changes in model behaviour as parameters are varied. Depending on the size of the underlying network, we observe oscillations and steady states in cell density exhibited in both models. Steady state behaviour can be described in large cell diffusion regimes via regular asymptotic expansions in the diffusion parameter. We numerically continue steady state solutions into intermediate diffusion regimes, where we observe symmetry-breaking bifurcations to both oscillatory and steady state behaviours that can be explained via local bifurcations, as well as symmetry-preserving oscillations that do not bifurcate from steady states. The spatially continuous analogue of the model only exhibits symmetric steady states and oscillatory solutions, and we conjecture that it is the finite lattice that gives rise to the more complicated symmetry-breaking dynamics. We suggest that the origin of both types of oscillations is a nonlocal reaction-diffusion mechanism mediated by quasi-static fluid flow. Finally we relate these results back to the original modelling question of how network topology influences tissue growth in a bioactive porous tissue scaffold.
• A. Krause, D. Beliaev, R. Van Gorder, S. Waters Lattice and Continuum Modelling of a Bioactive Porous Tissue Scaffold To appear in Mathematical Medicine and Biology. Available at: arXiv A contemporary procedure to grow artificial tissue is to seed cells onto a porous biomaterial scaffold and culture it within a perfusion bioreactor to facilitate the transport of nutrients to growing cells. Typical models of cell growth for tissue engineering applications make use of spatially homogeneous or spatially continuous equations to model cell growth, flow of culture medium, nutrient transport, and their interactions. The network structure of the physical porous scaffold is often incorporated in an averaged way through parameters in these models, either phenomenologically or through techniques like mathematical homogenization. We derive a model on a simple square grid lattice to demonstrate the importance of explicitly modelling the network structure of the porous scaffold, and compare results from this model with those from a modified continuum model from the literature. We capture two-way coupling between cell growth and fluid flow by allowing cells to block pores, and by allowing the shear stress of the fluid to affect cell growth and death. We explore a range of parameters for both models, and demonstrate quantitative and qualitative differences between predictions from each of these approaches, including spatial pattern formation on different timescales and local oscillations in cell density present only in the lattice model. These results suggest that for some parameter regimes, corresponding to specific cell types and scaffold geometries, the lattice model gives qualitatively different model predictions than typical continuum models.
• D. Beliaev, S. Muirhead. Discretisation schemes for level sets of planar Gaussian fields Communications in Mathematical Physics, May 2018, Volume 359, Issue 3, pp 869–913 Available at Springer: open access We study four discretisation schemes that extract information about level sets of planar Gaussian fields. Each scheme recovers information up to different level of precision, and each requires a maximum mesh-size in order to be valid with high probability. The first two schemes are generalisations and enhancements of similar schemes that have appeared in the literature; these give complete topological information about the level sets on either a local or global scale. As an application, we improve recent results on Russo-Seymour-Welsh estimates for the nodal set of positively-correlated planar Gaussian fields. The third and fourth schemes are, to the best of our knowledge, completely new. The third scheme is specific to the nodal set of the random plane wave, and provides global topological information about the nodal set up to visible ambiguities'. The fourth scheme gives an approximation of the Nazarov-Sodin constant for planar Gaussian fields.
• D. Beliaev, I. Wigman. Volume distribution of nodal domains of random band-limited functions, Probability Theory and Related Fields. Available at Springer: open access We study the volume distribution of nodal domains of random band-limited functions on generic manifolds, and find that in the high energy limit a typical instance obeys a deterministic universal law, independent of the manifold. Some of the basic qualitative properties of this law, such as its support, monotonicity and continuity of the cumulative probability function, are established.
• D. Grebenkov and D. Beliaev How anisotropy beats fractality in two-dimensional on-lattice DLA growth , Physical Review E. October 2017, Vol. 96(4) 042159. Published version. ArXiv version. We study the fractal structure of Diffusion-Limited Aggregation (DLA) clusters on the square lattice by extensive numerical simulations (with clusters having up to 108 particles). We observe that DLA clusters undergo strongly anisotropic growth, with the maximal growth rate along the axes. The naive scaling limit of a DLA cluster by its diameter is thus deterministic and one-dimensional. At the same time, on all scales from the particle size to the size of the entire cluster it has non-trivial box-counting fractal dimension which corresponds to the overall growth rate which, in turn, is smaller than the growth rate along the axes. This suggests that the fractal nature of the lattice DLA should be understood in terms of fluctuations around one-dimensional backbone of the cluster.
• D. Beliaev, V. Cammarota, I. Wigman Two point function for critical points of a random plane wave To appear in International Mathematical Research Notices. Available at OUP: open access Random plane wave is conjectured to be a universal model for high-energy eigenfunctions of the Laplace operator on generic compact Riemanian manifolds. This is known to be true on average. In the present paper we discuss one of important geometric observable: critical points. We first compute one-point function for the critical point process, in particular we compute the expected number of critical points inside any open set. After that we compute the short-range asymptotic behaviour of the two-point function. This gives an unexpected result that the second factorial moment of the number of critical points in a small disc scales as the fourth power of the radius.
• D. Beliaev, B. Duplantier, and M. Zinsmeister. Integral means spectrum of whole-plane SLE Communications in Mathematical Physics. July 2017, Volume 353, Issue 1, pages 119–133. Available at Springer: open access We complete the mathematical analysis of the fine structure of harmonic measure on SLE curves that was initiated by Beliaev and Smirnov, as described by the averaged integral means spectrum. For the unbounded version of whole-plane SLE as studied by Duplantier, Nguyen, Nguyen and Zinsmeister, and Loutsenko and Yermolayeva, a phase transition has been shown to occur for high enough moments from the bulk spectrum towards a novel spectrum related to the point at infinity. For the bounded version of whole-plane SLE studied here, a similar transition phenomenon, now associated with the SLE origin, is proved to exist for low enough moments, but we show that it is superseded by the earlier occurrence of the transition to the SLE tip spectrum.
• D. Grebenkov, D. Beliaev, and P. Jones. A Multiscale Guide to Brownian Motion, Journal of Physics A: Mathematical and Theoretical. 49 (2015) 043001 Available at: arXiv

Abstract: We revise the Levy's construction of Brownian motion as a simple though rigorous approach to operate with various Gaussian processes. A Brownian path is explicitly constructed as a linear combination of wavelet-based "geometrical features" at multiple length scales with random weights. Such a wavelet representation gives a closed formula mapping of the unit interval onto the functional space of Brownian paths. This formula elucidates many classical results about Brownian motion (e.g., non-differentiability of its path), providing intuitive feeling for non-mathematicians. The illustrative character of the wavelet representation, along with the simple structure of the underlying probability space, is different from the usual presentation of most classical textbooks. Similar concepts are discussed for Brownian bridge, fractional Brownian motion, Ornstein-Uhlenbeck process, Gaussian free field, and fractional Gaussian fields. Wavelet representations and dyadic decompositions form the basis of many highly efficient numerical methods to simulate Gaussian processes and fields, including Brownian motion and other diffusive processes in confining domains.

• D.Beliaev, Z. Kereta, On Bogomolny-Schmit conjecture, Journal of Physics A: Mathematical and Theoretical, November 2013 46.45 (2013): 455003 available at: arXiv

Abstract: Bogomolny and Schmit proposed that the critical edge percolation on the square lattice is a good model for the nodal domains of a random plane wave. Based on this they made a conjecture about the number of nodal domains. Recent computer experiments showed that the mean number of clusters per vertex and the mean number of nodal domains per unit area are very close but different. Since the original argument was mostly supported by numerics, it was believed that the percolation model is wrong. In this paper we give some numerical evidence in favour of the percolation model.

• D.Beliaev, F.Johansson Viklund, Some remarks on SLE bubbles and Schramm's two-point observable, Communications in Mathematical Physics, June 2013, Volume 320, Issue 2, pp 379-394, available at: arXiv

Abstract: Simmons and Cardy recently predicted a formula for the probability that the chordal SLE(8/3) path passes to the left of two points in the upper half-plane. In this paper we give a rigorous proof of their formula. Starting from this result, we derive explicit expressions for several natural connectivity functions for SLE(8/3) bubbles conditioned to be of macroscopic size. By passing to a limit with such a bubble we construct a certain chordal restriction measure and in this way obtain a proof of a formula for the probability that two given points are between two commuting SLE(8/3) paths. The one-point version of this result has been predicted by Gamsa and Cardy. Finally, we derive an integral formula for the second moment of the area of an SLE(8/3) bubble conditioned to have radius 1. We evaluate the area integral numerically and relate its value to a hypothesis that the area follows the Airy distribution.

• D.Beliaev, K.Izyurov, A proof of factorization formula for critical percolation, Communications in Mathematical physics. 310 (2012), no. 3, 611-623, available at: arXiv

Abstract: We give mathematical proofs to a number of statements which appeared in the series of papers by Kleban, Simmons and Ziff where they computed the probabilities of several percolation crossing events.

• D.Beliaev, S.Smirnov, Random Conformal Snowflakes, Annals of Mathematics 172 (2010), 597--615. PDF file.

Abstract: In many problems of classical analysis extremal configurations appear to exhibit complicated fractal structure, making it much harder to describe them and to attack such problems. This is particularly true for questions related to the multifractal analysis of harmonic measure.

We argue that, searching for extremals in such problems, one should work with random fractals rather than deterministic ones. We introduce a new class of fractals: random conformal snowflakes, and investigate its properties, developing tools to estimate spectra and showing that extremals can be found in this class. As an application we significantly improve known estimates from below on the extremal behaviour of harmonic measure, showing how to construct a rather simple snowflake, which has a spectrum quite close to the conjectured extremal value.

• D.Beliaev, E.Järvenpää, M.Järvenpää, A.Käenmäki, T.Rajala, S.Smirnov, V.Suomala, Packing dimension of mean porous measures, J. Lond. Math. Soc. (2) 80 (2009), no. 2, 514--530.,available at: arXiv

Abstract: We prove that the packing dimension of any mean porous Radon measure on ${\mathbb R}^n$ may be estimated from above by a function which depends on mean porosity. The upper bound tends to $d-1$ as mean porosity tends to its maximum value. Quite surprisingly, it turns out that mean porous measures are not necessarily approximable by mean porous sets. We verify this by constructing an example of a mean porous measure $\mu$ on $\mathbb R$ such that $\mu(A)=0$ for all mean porous sets $A \subset \mathbb R$.

• D.Beliaev, S.Smirnov, Harmonic measure and SLE. Comm. Math. Phys. 290 (2009), no. 2, 577--595. PDF file.

Abstract: The motivation for this paper is twofold: to study multifractal spectrum of the harmonic measure and to better describe the geometry of Schramm's SLE curves. Our main result is the rigorous computation the average spectrum of harmonic measure on domains bounded by SLE curves.

The average integral means spectrum $\bar\beta(t)$ of SLE$_\kappa$ is equal to

\begin{aligned} -t+\kappa\frac{4+\kappa-\sqrt{(4+\kappa)^2-8t\kappa}}{4\kappa} & \qquad t \le -1-\frac{3\kappa}{8}, \\ -t+\frac{(4+\kappa)(4+\kappa-\sqrt{(4+\kappa)^2-8t\kappa}\ )}{4\kappa} & \qquad -1-\frac{3\kappa}{8}\le t \le \frac{3(4+\kappa)^2}{32\kappa}, \\ t-\frac{(4+\kappa)^2}{16\kappa} & \qquad t\ge \frac{3(4+\kappa)^2}{32\kappa}. \end{aligned}

• D.Beliaev, Integral means spectrum of random conformal snowflakes, Nonlinearity 21 (2008), no. 7, 1435--1442. PDF file.

Abstract: In this paper we construct random conformal snowflakes with large integral means spectrum at different points. These new estimates are significant improvement over previously known lower bound of the universal spectrum. Our estimates are within 5-10 percent from the conjectured value of the universal spectrum.

• D.Beliaev, S.Smirnov, Harmonic measure on fractal sets, European Congress of Mathematics, 41--59, Eur. Math. Soc., Zürich, 2005. PDF file.

Abstract: Many problems in complex analysis can be reduced to the evaluation of the universal spectrum: the supremum of multifractal spectra of harmonic measures for all planar domains. Its exact value is still unknown, with very few estimates available. We start with a brief survey of related problems and available estimates from above. Then we discuss in more detail estimates from below, describing the search for a fractal domain which attains the maximal possible spectrum

• D.Beliaev, S.Smirnov, On Littlewood's constants, Bull. London Math. Soc. 37 (2005), no. 5, 719--726. PDF file.

Abstract: In two papers, Littlewood studied seemingly unrelated constants: (i) the best $\alpha$ such that for any polynomial $f$ of degree $n$ the areal integral of its spherical derivative is at most $\mathrm{ const}\cdot n^\alpha$ and (ii) the extremal growth rate $\beta$ of the length of Green’s equipotentials for simply connected domains. These two constants are shown to coincide, thus greatly improving known estimates on $\alpha$.

• A.Aleman, D.Beliaev, and H.Hedenmalm, Real zero polynomials and Pólya-Schur type theorems, Journal d'Analyse Mathématique, 94 (2004), 49--60. PDF file.
• D.Beliaev, S.Smirnov, On dimension of porous measures, Math. Ann. 323 (2002), no. 1, 123--141. PDF file.
• D.Beliaev, V.Havin, On the uncertainty principle for M. Riesz potentials, Ark. Mat. 39 (2001), no. 2, 223--243. PDF file.

Simulations

Here you can find some simulations that I have made in the recent years. You are welcome to use them for any purpose as long as they are properly attributed. I would appreciate if you let me know if you use any of these figures. You are welcome to contact me if you need any of these pictures in higher resolution or need any similar pictures.