Dmitry Beliaev

Harmonic Measure and SLE

In this paper we study the multifractal structure of Schramm’s SLE curves. We derive the values of the (average) spectrum of harmonic measure and prove Duplantier’s prediction for the multifractal spectrum of SLE curves. The spectrum can also be used to derive estimates of the dimension, Hölder exponent and other geometrical quantities. The SLE curves provide perhaps the only example of sets where the spectrum is non-trivial yet exactly computable.

Packing dimension of mean porous measures

We prove that the packing dimension of any mean porous Radon measure on Rd 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. This result was stated in D. B. Beliaev and S. K. Smirnov [�eOn dimension of porous measures�f, Math. Ann. 323 (2002) 123.141], and in a weaker form in E. J�Narvenp�Na�Na and M. J�Narvenp�Na�Na [�ePorous measures on Rn: local structure and dimensional properties�f, Proc. Amer. Math. Soc. (2) 130 (2002) 419.426], but the proofs are not correct. 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 �E on R such that �E(A) = 0 for all mean porous sets A �¼ R.

Real zero polynomials and Polya-Schur type theorems

Real zero preserving operators. Let P(C) denote the space of all polynomials with complex coefficients, regarded as functions on the complex plane. The differentiation operator D = d/dz acts on P(C); the action on the monomials is given by D z = n zn−1 for n = 1, 2, 3, . . .. We also have the operator D∗ of multiplicative differentiation, related toD viaD∗ = zD; the action on the monomials is given byD∗ z = n z for n = 0, 1, 2, . . .. Whereas D commutes with all translations, D∗ commutes with all dilations of the complex plane C. A third differentiation operator D] = zD is of interest; its action on the monomials is given by D] z = n z for n = 0, 1, 2, . . .. We get it by first inverting the plane (z 7→ 1/z), then applying minus differentiation −D, and by finally inverting back again. This means that studying D] on the polynomials is equivalent to studying the ordinary differentiation operator D on the space of all rational functions that are regular at all points of the extended plane with the exception of the origin. The Gauss-Lucas theorem states that if a polynomial p(z) has its zeros contained in some given convex set K, then its derivative Dp(z) = p′(z) has all its zeros in K as well (unless p(z) is constant, that is). In particular, if all the zeros are real, then so are the zeros of the derivative. Naturally, the same statement can be made for the multiplicative derivative D∗ as well, and for D], too. In this context, we should mention the classical theorem of Laguerre [2, p. 23], which extends the Gauss-Lucas theorem for the real zeros case to the more general setting of entire functions of genus 0 or 1. To simplify the later discussion, we introduce the notation P(C;R) for the collection of all polynomials with only real zeros, including all constants. This means that the zero polynomial is in P(C;R), although strictly speaking, it has plenty of non-real zeros. Clearly, P(C;R) constitutes a multiplicative semi-group. Let T : P(C)→ P(C) be a linear operator. Let us say that T is real zero preserving if T (P(C;R)) ⊂ P(C;R); it would be of interest to have a complete characterization of the real zero preserving operators. From the above remarks, we know that D, D∗, and D] are real zero preserving. To get some headway into this general problem, it is helpful to have some additional information regarding the given operator T .

Volume distribution of nodal domains of random band-limited functions

We study the volume distribution of nodal domains of families of naturally arising Gaussian random fields on generic manifolds, namely random band-limited functions. It is found 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.

Discretisation Schemes for Level Sets of Planar Gaussian Fields

Smooth random Gaussian functions play an important role in mathematical physics, a main example being the random plane wave model conjectured by Berry to give a universal description of high-energy eigenfunctions of the Laplacian on generic compact manifolds. Our work is motivated by questions about the geometry of such random functions, in particular relating to the structure of their nodal and level sets. We study four discretisation schemes that extract information about level sets of planar Gaussian fields. Each scheme recovers information up to a 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 (Beffara and Gayet in Publ Math IHES, 2017. https://doi.org/10.1007/s10240-017-0093-0; Mischaikow and Wanner in Ann Appl Probab 17:980–1018, 2007); these give complete topological information about the level sets on either a local or global scale. As an application, we improve the results in Beffara and Gayet (2017) 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 a way to approximate the mean number of excursion domains of planar Gaussian fields.

Integral means spectrum of random conformal snowflakes

It is known that the multifractal spectrum of harmonic measure plays a central role in the geometric function theory. Recently Smirnov and Beliaev introduced a new class of random fractals that are called random conformal snowflakes. They proved that the multifractal spectrum of snowflakes is related to the spectral radius of a particular integral operator. In this paper we exploit this connection to estimate the multifractal spectrum of snowflakes.

How nanoscale protein interactions determine the mesoscale dynamic organisation of bacterial outer membrane proteins

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.In Escherichia coli, outer membrane protein (OMP) cluster and form islands, but the origin and behaviour of those clusters remains poorly understood. Here authors use coarse grained molecular dynamics simulation and show that their mesoscale simulations recapitulate the restricted diffusion characteristics of OMPs.

Lattice and continuum modelling of a bioactive porous tissue scaffold

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 through parameters in these models, either phenomenologically or through techniques like mathematical homogenization. We derive a model on a 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 and local oscillations in cell density present only in the lattice model. These differences 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. Our results inform model selection for bioactive porous tissue scaffolds, aiding in the development of successful tissue engineering experiments and eventually clinically successful technologies.

How anisotropy beats fractality in two-dimensional on-lattice diffusion-limited-aggregation growth

We study the fractal structure of diffusion-limited aggregation (DLA) clusters on a square lattice by extensive numerical simulations (with clusters having up to 10^{8} 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 a nontrivial 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 the one-dimensional backbone of the cluster.