Related papers: Internal aggregation models with multiple sources …
Some families of graphs, such as the n-cubes and Sierpinski gaskets, are self-similar. In this paper we show how such recursive structure can be used systematically to prove isoperimetric theorems.
In the classical model of Diffusion Limited Aggregation (DLA), introduced by Witten and Sander, the process begins with a single particle cluster placed at the origin of a space, and then, one at a time, particles make a random walk from…
We analyze the ground state structure of the supersymmetric sine-Gordon model via the lattice regularization. The nonlinear integral equations are derived for any values of the boundary parameters by the analytic continuation and showed…
We study the number of connected spanning subgraphs $f_{d,b}(n)$ on the generalized Sierpinski gasket $SG_{d,b}(n)$ at stage $n$ with dimension $d$ equal to two, three and four for $b=2$, and layer $b$ equal to three and four for $d=2$. The…
This thesis is about local and non-local Dirichlet forms on the Sierpi\'nski gasket and the Sierpi\'nski carpet. We are concerned with the following three problems in analysis on the Sierpi\'nski gasket and the Sierpi\'nski carpet. First, a…
We consider a stochastic aggregation model on Z^d. Start with particles located at the vertices of the lattice, initially distributed according to the product Bernoulli measure with parameter \mu. In addition, there is an aggregate, which…
Following the recent work of Sznitman (arXiv:0805.4516), we investigate the microscopic picture induced by a random walk trajectory on a cylinder of the form G_N x Z, where G_N is a large finite connected weighted graph, and relate it to…
The divisible sandpile model is a fixed-energy continuous counterpart of the Abelian sandpile model. We start with a random initial configuration and redistribute mass deterministically. Under certain conditions the sandpile will stabilize.…
In the Diffusion Limited Aggregation (DLA) process on on $\mathbb{Z}^2$, or more generally $\mathbb{Z}^d$, particles aggregate to an initially occupied origin by arrivals on a random walk. The scaling limit of the result, empirically, is a…
We consider (a variant of) the external multi-particle diffusion-limited aggregation (MDLA) process of Rosenstock and Marquardt on the plane. Based on the recent findings of [11], [10] in one space dimension it is natural to conjecture that…
The distributions of $ N $-particle systems of Gaussian unitary ensembles converge to Sine$_2$ point processes under bulk-scaling limits. These scalings are parameterized by a macro-position $ \theta $ in the support of the semicircle…
Start with a graph with a subset of vertices called {\it the border}. A particle released from the origin performs a random walk on the graph until it comes to the immediate neighbourhood of the border, at which point it joins this subset…
Given finite configurations $P_1, \dots, P_n \subset \mathbb{R}^d$, let us denote by $\mathbf{m}_{\mathbb{R}^d}(P_1, \dots, P_n)$ the maximum density a set $A \subseteq \mathbb{R}^d$ can have without containing congruent copies of any…
For $n\in\mathbb{N}$, let $\{X^n_i\}$ be an infinite collection of Brownian particles on the real line where the leftmost particle $\min_iX^n_i(t)$ is given a drift $n$, and let $\mu^n_t=n^{-1}\sum_i\delta_{X^n_i(t)}$, $t\ge0$ denote the…
In this paper we consider a class of obstacle problems of the type %\begin{equation*} %\int_{\Omega}\left<A(x, Du), D(\varphi-u)\right> \, \dx\ge0\qquad\forall %\varphi\in W^{1,q}(\Omega) \quad {\mathrm{s.t.}} \quad \varphi \ge \psi…
We consider an infinite system of coupled stochastic differential equations (SDE) describing dynamics of the following infinite particle system. Each partricle is characterised by its position $x\in \mathbb{R}^{d}$ and internal parameter…
A Skeleton-stabilized IsoGeometric Analysis (SIGA) technique is proposed for incompressible viscous flow problems with moderate Reynolds number. The proposed method allows utilizing identical finite dimensional spaces (with arbitrary…
Diffusion-Limited Aggregation (DLA) is a cluster-growth model that consists in a set of particles that are sequentially aggregated over a two-dimensional grid. In this paper, we introduce a biased version of the DLA model, in which…
In two dimensions, the $l$-level Sierpinski gasket $\mathrm{SG}(l)$ is obtained by splitting an equilateral triangle into a collection of $l^2$ equilateral triangles of equal size and with the same total area, retaining only the $l(l+1)/2$…
We consider an infinite system of particles on the positive real line, initiated from a Poisson point process, which move according to Brownian motion up until the hitting time of a barrier. The barrier increases when it is hit, allowing…