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

MT22: Lecturer for Metric Spaces and Complex Analysis. Lecture notes and problem sheets are available at course page

HT23: Lecturer for B8.2 Continuous Martingales

Projects and Theses

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

This year I am not supervising Part C/OMMS dissertations

Postdocs

• Stephen Muirhead. Postdoc at Oxford 2015-2017. Currently a DECRA Fellow at the University of Melbourne.
• Riccardo Maffucci. Postdoc at Oxford 2017-2019. Currently a Lecturer in graph theory, EPFL, Switzerland.

Graduate students

• Akshay Hegde. Started 2021.
• Michael McAuley. DPhil 2020. Joint supervision with S. Muirhead. Thesis "Excursion sets of planar Gaussian fields".
• Vlad Margarint. DPhil 2019. Joint supervision with Prof. T. Lyons. Thesis "Pathwise and probabilistic analysis in the context of Schramm-Loewner Evolutions (SLE)".
• Andrew Krause. DPhil 2017. Joint supervision with Prof. S. Waters and Dr. R. Van Gorder. Thesis "Network Modelling of Bioactive Porous Media".

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".

Books

• D. Beliaev Conformal Maps and Geometry (Advanced Textbooks in Mathematics) is available from World Scientific and Amazon

Preprints

• D. Beliaev, Gaussian fields and percolation Available at: arXiv
  Abstract In the last two decades there was a lot of progress in understanding the geometry of smooth Gaussian fields. This survey aims to cover one particular line of research: the large scale behaviour of level and excursion sets and their (conjectured) connection to the percolation theory.
• D. Beliaev, M. McAuley, S. Muirhead A central limit theorem for the number of excursion set components of Gaussian fields Available at: arXiv
  Abstract For a smooth stationary Gaussian field on ${\mathbb{R}}^d$ and level $\ell\in \mathbb{R}$, we consider the number of connected components of the excursion set $\{f\ge \ell\}$ (or level set $\{f=\ell\}$) contained in large domains. The mean of this quantity is known to scale like the volume of the domain under general assumptions on the field. We prove that, assuming sufficient decay of correlations (e.g. the Bargmann-Fock field), a central limit theorem holds with volume-order scaling. Previously such a result had only been established for `additive' geometric functionals of the excursion/level sets (e.g. the volume or Euler characteristic) using Hermite expansions. Our approach, based on a martingale analysis, is more robust and can be generalised to a wider class of topological functionals. A major ingredient in the proof is a third moment bound on critical points, which is of independent interest.
• D. Beliaev, T. Lyons, V. Margarint A new approach to SLE phase transition Available at: arXiv
  Abstract It is well know that SLE($\kappa$) curves exhibit a phase transition at $\kappa=4$. For $\kappa\le 4$ they are simple curves with probability one, for $\kappa>4$ they are not. The standard proof is based on the analysis of the Bessel SDE of dimension $d=1+4/\kappa$. We propose a different approach which is based on the analysis of the Bessel SDE with $d=1-4/\kappa$. This not only gives a new perspective, but also allows to describe the formation of the SLE `bubbles' for $\kappa>4$.

Published and accepted papers

• D. Beliaev, A. Shekhar, V. Margarint Continuity of Zero-Hitting Times of Bessel Processes and Welding Homeomorphisms of SLE ${}_\kappa$ Latin American Journal of Probability and Mathematical Statistics. Open access. Available at: arXiv
  Abstract We consider a family of Bessel Processes that depend on the starting point $x$ and dimension $\delta$, but are driven by the same Brownian motion. Our main result is that almost surely the first time a process hits 0 is jointly continuous in $x$ and $\delta$, provided $\delta\le 0$. As an application, we show that the SLE($\kappa$) welding homeomorphism is continuous in $\kappa$ for $\kappa\in[0,4]$. Our motivation behind this is to study the well known problem of the continuity of SLE($\kappa$) in $\kappa$. The main tool in our proofs is random walks with increments distributed as infinite mean Inverse-Gamma laws.
• D. Beliaev, V. Cammarota, I. Wigman No repulsion between critical points for planar Gaussian random fields Electronic Communications in Probability. Open access. Available at: arXiv
  Abstract We study the behaviour the point process of critical points of isotropic stationary Gaussian fields. We compute the main term in the asymptotic expansion of the two-point correlation function near the diagonal. Our main result could be interpreted as a statement that for a 'generic' field the critical points neither repel no attract each other. Our analysis also allows to study how the short-range behaviour of critical points depends on their index.
• D. Beliaev, R. Maffucci Coupling of stationary fields with application to arithmetic waves Stochastic Processes and their Applications. Open access. Available at: arXiv
  Abstract In this paper we obtain a range of quantitative results of the following type: given two centered Gaussian fields with close covariance kernels we construct a coupling such that the fields are uniformly close on some compact with probability very close to one. As an application, we show that it is possible to couple arithmetic random waves so that they converge locally uniformly to the random plane wave and estimate the rate of convergence.
• D. Beliaev, M. McAuley, S. Muirhead Fluctuations of the number of excursion sets of planar Gaussian field Probability and Mathematical Physics. Open access. Available at: arXiv
  Abstract The number of connected components of the excursion set above a level $\ell$ (or level set at $\ell$) of a smooth planar Gaussian field in the ball of radius $R$ is known to have mean of order $R^2$ for any $\ell$. We show that for certain fields with positive spectral density near the origin (including the Bargmann-Fock field), and for certain values of $\ell$, these random variables have fluctuations of order at least $R$, and hence, variance of order at least $R^2$. In particular this holds for excursion sets when $\ell$ is in some neighbourhood of zero, and it holds for excursion/level sets when $|\ell|$ is sufficiently large. We prove stronger fluctuation lower bounds of order $R^\alpha$, $\alpha\in[1,2]$, in the case that the spectral density has a singularity at the origin. Finally we show that the number of excursion/level sets for the random plane wave at certain levels has fluctuations of order at least $R^{3/2}$ and hence variance of order at least $R^{3}$. We expect that these bounds are of the correct order, at least for generic levels.
• D. Beliaev, R. Maffucci Intermediate and small scale limiting theorems for random fields Communications in Number Theory and Physics. Published version. Available at: arXiv
  Abstract In this paper we study the nodal lines of random eigenfunctions of the Laplacian on the torus, the so called 'arithmetic waves'. To be more precise, we study the number of intersections of the nodal line with a straight interval in a given direction. We are interested in how this number depends on the length and direction of the interval and the distribution of spectral measure of the random wave. We analyse the second factorial moment in the short interval regime and the persistence probability in the long interval regime. We also study relations between the Cilleruelo and Cilleruelo-type fields. We give an explicit coupling between these fields which on mesoscopic scales preserves the structure of the nodal sets with probability close to one.
• D. Beliaev, S. Muirhead, I. Wigman Mean conservation of nodal volume and connectivity measures for Gaussian ensembles Advances in Mathematics. Open access. Available at: arXiv
  Abstract We study in depth the nesting graph and volume distribution of the nodal domains of a Gaussian field, which have been shown in previous works to exhibit asymptotic laws. A striking link is established between the asymptotic mean connectivity of a nodal domain (ie the vertex degree in its nesting graph) and the positivity of the percolation probability of the field, along with a direct dependence of the average nodal volume on the percolation probability. Our results support the prevailing ansatz that the mean connectivity and volume of a nodal domain is conserved for generic random fields in dimension $d=2$ but not in $d\ge 3$ , and are applied to a number of concrete motivating examples.
• D. Beliaev, S. Muirhead, I. Wigman Russo-Seymour-Welsh estimates for the Kostlan ensemble of random polynomials Annales de l'Institut Henri Poincaré, Probabilités et Statistiques. Published version. Available at: arXiv
  Abstract

  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.
• D. Beliaev, M. McAuley, S. Muirhead Smoothness and monotonicity of the excursion set density of planar Gaussian fields Electronic Journal of Probability. Open access. Available at: arXiv
  Abstract Nazarov and Sodin have shown that the number of connected components of the nodal set of a planar Gaussian field in a ball of radius $R$, normalised by area, converges to a constant. This has been generalised to excursion/level sets at arbitrary levels, implying the existence of functionals $c_{ES}(\ell)$ and $c_{LS}(\ell)$ that encode the density of excursion/level set components at the level $\ell$. We prove that these functionals are continuously differentiable for a wide class of fields. This follows from a more general result which derives differentiability of the functionals from the decay of the probability of `four-arm events' for the field conditioned to have a saddle point at the origin. For some fields, including the important special cases of the Random Plane Wave and the Bargmann-Fock field, we also derive stochastic monotonicity of the conditioned field, which allows us to deduce regions on which $c_{ES}(\ell)$ and $c_{LS}(\ell)$ are monotone.
• D. Beliaev, T. Lyons, V. Margarint Continuity in κ in SLEκ theory using a constructive method and Rough Path Theory Annales de l'Institut Henri Poincaré, Probabilités et Statistiques. Published version. Available at: arXiv
  Abstract Questions regarding the continuity in $\kappa$ of the SLE($\kappa$) traces and maps appear very naturally in the study of SLE. In order to study the first question, we consider a natural coupling of SLE traces: for different values of $\kappa$ we use the same Brownian motion. It is very natural to assume that with probability one, SLE($\kappa$) depends continuously on $\kappa$. It is rather easy to show that SLE is continuous in the Carathéodory sense, but showing that SLE traces are continuous in the uniform sense is much harder. In this note we show that for a given sequence $\kappa_j \to \kappa\in (0,8/3)$, for almost every Brownian motion SLE($\kappa$) traces converge locally uniformly. This result was also recently obtained by Friz, Tran and Yuan using different methods. In our analysis, we provide a constructive way to study the SLEκ traces for varying parameter $\kappa \in (0,8/3)$. The argument is based on a new dynamical view on the approximation of SLE curves by curves driven by a piecewise square root approximation of the Brownian motion. The second question can be answered naturally in the framework of Rough Path Theory. Using this theory, we prove that the solutions of the backward Loewner Differential Equation driven by $\sqrt{\kappa}B_t$ when started away from the origin are continuous in the $p$-variation topology in the parameter$\kappa$, for all $\kappa\ge 0$.
• D. Beliaev, S. Muirhead, A. Rivera A covariance formula for topological events of smooth Gaussian fields Annals of Probability. Published version. Available at: arXiv
  Abstract We derive a covariance formula for the class oftopological events' of smooth Gaussian fields on manifolds; these are events that depend only on the topology of the level sets of the field, for example (i) crossing events for level or excursion sets,(ii) events measurable with respect to the number of connected components of level or excursion sets of a given diffeomorphism class, and (iii) persistence events. As an application of the covariance formula, we derive strong mixing bounds for topological events, as well as lower concentration inequalities for additive topological functionals (eg the number of connected components) of the level sets that satisfy a law of large numbers. The covariance formula also gives an alternate justification of the Harris criterion, which conjecturally describes the boundary of the percolation university class for level sets of stationary Gaussian fields. Our work is inspired by a recent paper by Rivera and Vanneuville, in which a correlation inequality was derived for certain topological events on the plane, as well as by an old result of Piterbarg, in which a similar covariance formula was established for finite-dimensional Gaussian vectors.
• D. Beliaev, M. McAuley, S. Muirhead On the number of excursion sets of planar Gaussian fields Probability Theory and Related Fields. Open access Available at: arXiv
  Abstract 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.
• 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
  Abstract 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 Bifurcations and Dynamics Emergent From Lattice and Continuum Models of Bioactive Porous Media International Journal of Bifurcation and Chaos Vol. 28, No. 11, 1830037 (2018). Published version. Available at: arXiv
  Abstract 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 Mathematical Medicine and Biology: A Journal of the IMA, Volume 36, Issue 3, September 2019, Pages 325–360. Published version. Available at: arXiv
  Abstract 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
  Abstract 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
  Abstract 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.
  Abstract 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 International Mathematics Research Notices, Volume 2019, Issue 9, May 2019, Pages 2661–2689. Available at OUP: open access
  Abstract 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
  Abstract 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.
• 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.
  The original publication is available on Springer's LINK at http://link.springer.de/...
• 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.

Small scale random plane wave sample

Nodal domains of a random plane wave. Download: PNG or PDF

Nodal domains and critical points. Download: PNG or PDF

Nodal domains and gradient flow graph. Download: PNG or PDF

Gradient flow graph. Download: PNG or PDF

Nodal domains and percolation on the gradient flow graph. Download: PNG or PDF

Large scale sample of critical points of a random plane wave

Critical points. Download: PNG or PDF

Extrema only. Download: PNG or PDF

Gradient flow graph. Download: PNG or PDF

Spherical Gaussian fields

Random spherical harmonic. Download in high resolution: PNG

Random band-limited function. Download in high resolution: PNG

Kostlan's ensemble of degree 300. Download in high resolution: PNG

Research Group and Seminars

• I am a member of the Stochastic Analysis group.
• Stochastic Analysis & Mathematical Finance Seminars are on Mondays at 14.15 and 15.45. These are held in the Mathematical Institute. The seminar schedule is also available as a public Google calendar.
• Informal Probability Workshop is on Monday noon. The talks are in Mathematical Institute