Related papers: Discrete Growth Models
We investigate the function $d_\mathbf{A}(n)$, which gives the size of a least size generating set for $\mathbf{A}^n$, in the case where $\mathbf{A}$ has a cube term. We show that if $\mathbf{A}$ has a $k$-cube term and $\mathbf{A}^k$ is…
Models of black holes in (1+1)-dimensions provide a theoretical laboratory for the study of semi-classical effects of realistic black holes in Einstein's theory. Important examples of two-dimensional models are given by string theory…
We show that there are two classes of solutions that describe static spherically symmetric dyonic dilaton black holes with two nonsingular horizons. The first class includes only the already known solutions that exist for a few special…
We investigate the evolutionary history of the Universe's metal content focusing on the chemical abundance of several elements (N, O, S, Si, Fe, Cr, Zn) taken from observational data and predictions from chemical evolution models. The…
The original 2017 version of this paper, published in Ann. Appl. Probab., 27, 1678--1801, contains a major gap in the proofs. In the subsequent publication in Ann. Appl. Probab., 34, 3370--3374, 2024, we indicated how to fix this. For…
We investigate the formation of an infinite cluster of entangled threads in a (2+1)-dimensional system. We demonstrate that topological percolation belongs to the universality class of the standard 2D bond percolation. We compute the…
We consider the DLA process on a cylinder G x N. It is shown that this process "grows arms", provided that the base graph G has small enough mixing time. Specifically, if the mixing time of G is at most (log|G|)^(2-\eps), the time it takes…
Forward models of the galaxy density field enable simulation based inference as well as field level inference of galaxy clustering. However, these analysis techniques require forward models that are both computationally fast and robust to…
The evolution of structure in biology is driven by accretion and change. Accretion brings together disparate parts to form bigger wholes. Change provides opportunities for growth and innovation. Here we review patterns and processes that…
Lie symmetry algebra of the dispersionless Davey-Stewartson (dDS) system is shown to be infinite-dimensional. The structure of the algebra turns out to be Kac-Moody-Virasoro one, which is typical for integrable evolution equations in…
We consider internal diffusion limited aggregation in dimension larger than or equal to two. This is a random cluster growth model, where random walks start at the origin of the d-dimensional lattice, one at a time, and stop moving when…
Developments in dynamical systems theory provides new support for the discretisation of \pde{}s and other microscale systems. By systematically resolving subgrid microscale dynamics the new approach constructs asymptotically accurate,…
The article is part of a review volume on the formation of the first black holes and summarises FLRW-cosmologies, the statistical description of cosmic structures as Gaussian random fields, as well as fluid mechanics in the linear and…
For a natural number $N\geq 2$ and a real $\alpha$ such that $0 < \alpha \leq \sqrt{N}-1$, we define $I_\alpha:=[\alpha,\alpha+1]$ and $I_\alpha^-:=[\alpha,\alpha+1)$ and investigate the continued fraction map $T_\alpha:I_\alpha \to…
A method of constructing (finitely generated and projective) right module structure on a finitely generated projective left module over an algebra is presented. This leads to a construction of a first order differential calculus on such a…
We study the black hole information paradox in the context of a two-dimensional toy model given by dilaton gravity coupled to $N$ massless scalar fields. After making the model well-defined by imposing reflecting boundary conditions at a…
A new model of Laplacian stochastic growth is formulated using conformal mappings. The model describes two growth regimes, stable and turbulent, separated by a sharp phase transition. The first few Fourier components of the mapping define…
While persistent homology has taken strides towards becoming a wide-spread tool for data analysis, multidimensional persistence has proven more difficult to apply. One reason is the serious drawback of no longer having a concise and…
We study two variants of the fundamental problem of finding a cluster in incomplete data. In the problems under consideration, we are given a multiset of incomplete $d$-dimensional vectors over the binary domain and integers $k$ and $r$,…
The current paper is a short review of rigorous results for the 1-2 model. The 1-2 model on the hexagonal lattice is a model of statistical mechanics in which each vertex is constrained to have degree either 1 or 2. It was proposed in a…