Related papers: A shape theorem and semi-infinite geodesics for th…
We study the behavior of the random walk in a continuum independent long-range percolation model, in which two given vertices $x$ and $y$ are connected with probability that asymptotically behaves like $|x-y|^{-\alpha}$ with $\alpha>d$,…
In the course of classifying the homogeneous permutations, Cameron introduced the viewpoint of permutations as structures in a language of two linear orders, and this structural viewpoint is taken up here. The majority of this thesis is…
This doctoral thesis undertakes an in-depth exploration of limiting shape theorems across diverse mathematical structures, with a specific focus on subadditive processes within finitely generated groups exhibiting polynomial growth rates,…
In this paper we provide general conditions on a one parameter family of random infinite subsets of Z^d to contain a unique infinite connected component for which the chemical distances are comparable to the Euclidean distances, focusing…
The third part of the paper concludes the proof of the main result --- the description of the ergodic decomposition of infinite Pickrell measures. First it is shown that the scaling limit of radial parts of finite-dimensional infinite…
We present a new technique to apply finite element methods to partial differential equations over curved domains. A change of variables along a coordinate transformation satisfying only low regularity assumptions can translate a Poisson…
Last passage percolation (LPP) is a model of a directed metric and a zero-temperature polymer where the main observable is a directed path evolving in a random environment accruing as energy the sum of the random weights along itself. When…
It is believed that, under very general conditions, bi-infinite geodesics (or bigeodesics) do not exist for planar first and last passage percolation (LPP) models. However, if one endows the model with a natural dynamics, thereby gradually…
The main purpose of this paper is to introduce and establish basic results of a natural extension of the classical Boolean percolation model (also known as the Gilbert disc model). We replace the balls of that model by a positive…
This article discusses a relatively new geometric flow, called the hypersymplectic flow. In the first half of the article we explain the original motivating ideas for the flow, coming from both 4-dimensional symplectic topology and…
Considered are the large $N$, or large intensity, forms of the distribution of the length of the longest increasing subsequences for various models. Earlier work has established that after centring and scaling, the limit laws for these…
In this paper we study Poisson processes of so-called "fat" cylinders in hyperbolic space. As our main result we show that this model undergoes a percolation phase transition. We prove this by establishing a novel link between the fat…
We study a symmetrized (half-space) version of geometric last passage percolation with a boundary parameter $c$ that interpolates between subcritical, critical, and supercritical behavior. This model gives rise to a family of interlacing…
We study the two-time distribution in directed last passage percolation with geometric weights in the first quadrant. We compute the scaling limit and show that it is given by a contour integral of a Fredholm determinant.
We study a version of first passage percolation on $\mathbb{Z}^d$ where the random passage times on the edges are replaced by contact times represented by random closed sets on $\mathbb{R}$. Similarly to the contact process without…
We consider the motion of a two-dimensional body of arbitrary shape in a planar irrotational, incompressible fluid with a given amount of circulation around the body. We derive the equations of motion for this system by performing…
We establish non-uniqueness regimes for the infinite-volume two-colored Widom--Rowlinson model based on inhomogeneous Poisson point processes with locally finite intensity measures featuring percolation. As an application, we provide…
The Sample Compression Conjecture of Littlestone & Warmuth has remained unsolved for over two decades. This paper presents a systematic geometric investigation of the compression of finite maximum concept classes. Simple arrangements of…
We show that classical integrable models of last passage percolation and the related nonintersecting random walks converge uniformly on compact sets to the Airy line ensemble. Our core approach is to show convergence of nonintersecting…
We introduce a model of a randomly growing interface in multidimensional Euclidean space. The growth model incorporates a random order model as an ingredient of its graphical construction, in a way that replicates the connection between the…