# 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
• D. Beliaev, M. McAuley, S. Muirhead A central limit theorem for the number of excursion set components of Gaussian fields Available at: arXiv
• D. Beliaev, T. Lyons, V. Margarint A new approach to SLE phase transition Available at: arXiv

## 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
• 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
• D. Beliaev, R. Maffucci Coupling of stationary fields with application to arithmetic waves Stochastic Processes and their Applications. Open access. Available at: arXiv
• 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
• D. Beliaev, R. Maffucci Intermediate and small scale limiting theorems for random fields Communications in Number Theory and Physics. Published version. Available at: arXiv
• 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
• 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
• 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
• 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
• D. Beliaev, S. Muirhead, A. Rivera A covariance formula for topological events of smooth Gaussian fields Annals of Probability. Published version. Available at: arXiv
• 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
• 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
• 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
• 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
• 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
• D. Beliaev, I. Wigman. Volume distribution of nodal domains of random band-limited functions, Probability Theory and Related Fields. Available at Springer: open access
• 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.
• 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
• 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
• 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
• D.Beliaev, Z. Kereta, On Bogomolny-Schmit conjecture, Journal of Physics A: Mathematical and Theoretical, November 2013 46.45 (2013): 455003 available at: arXiv
• 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
• 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
• D.Beliaev, S.Smirnov, Random Conformal Snowflakes, Annals of Mathematics 172 (2010), 597--615. PDF file.
• 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
• D.Beliaev, S.Smirnov, Harmonic measure and SLE. Comm. Math. Phys. 290 (2009), no. 2, 577--595. PDF file.
• D.Beliaev, Integral means spectrum of random conformal snowflakes, Nonlinearity 21 (2008), no. 7, 1435--1442. PDF file.
• D.Beliaev, S.Smirnov, Harmonic measure on fractal sets, European Congress of Mathematics, 41--59, Eur. Math. Soc., Zürich, 2005. PDF file.
• D.Beliaev, S.Smirnov, On Littlewood's constants, Bull. London Math. Soc. 37 (2005), no. 5, 719--726. PDF file.
• 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