相关论文: Convergence of uniform triangulations under the Ca…

Proximity maps and regions are defined based on the relative allocation of points from two or more classes in an area of interest and are used to construct random graphs called proximity catch digraphs (PCDs) which have applications in…

We introduce series-triangular graph embeddings and show how to partition point sets with them. This result is then used to improve the upper bound on the number of Steiner points needed to obtain compatible triangulations of point sets.…

Percolation on a plane is usually associated with clusters spanning two opposite sides of a rectangular system. Here we investigate three-leg clusters generated on a square lattice and spanning the three sides of equilateral triangles. If…

Two triangle meshes are conformally equivalent if for any pair of incident triangles the absolute values of the corresponding cross-ratios of the four vertices agree. Such a pair can be considered as preimage and image of a discrete…

We consider the strongly connected components (SCCs) of a uniform directed graph on $n$ vertices with i.i.d. in- and out-degree pairs distributed as $(D^-,D^+)$, with $\mathbb E[D^+]=\mathbb E[D^-]=\mu$. We condition on equal total in- and…

We consider the conformal mapping of the Bunimovich stadium, a region enclosed by a Jordan curve with four smooth corners, primarily in the context of a particle undergoing Brownian motion within its closed geometry with Dirichlet boundary…

In this paper, we will compute the dimension of the space of spun and ordinary normal surfaces in an ideal triangulation of the interior of a compact 3-manifold with incompressible tori or Klein bottle components. Spun normal surfaces have…

We prove that the metric space associated with a uniformly distributed planar quadrangulation with n faces and no pendant vertices converges modulo a suitable rescaling to the Brownian map. This is a first step towards the extension of…

Sheffield (2011) introduced an inventory accumulation model which encodes a random planar map decorated by a collection of loops sampled from the critical Fortuin-Kasteleyn (FK) model. He showed that a certain two-dimensional random walk…

For each $n\in\mathbb{N}$, let $\mathbf{Q}_n$ be a uniform rooted measured quadrangulation of size $n$ conditioned to have $r(n)$ vertices in its root block. We prove that for a suitable function $r(n)$, after rescaling graph distance by…

We revisit the 3-states Potts model on random planar triangulations as a Hermitian matrix model. As a novelty, we obtain an algebraic curve which encodes the partition function on the disc with both fixed and mixed spin boundary conditions.…

The aim of this paper is to give an upper bound for the intrinsic diameter of a surface with boundary immersed in a conformally flat three dimensional Riemannian manifold in terms of the integral of the mean curvature and of the length of…

The (non-spanning) tree-decorated quadrangulation is a random pair formed by a quadrangulation and a subtree chosen uniformly over the set of pairs with prescribed size. In this paper we study the tree-decorated quadrangulation in the…

We describe an efficient algorithm to compute a conformally equivalent metric for a discrete surface, possibly with boundary, exhibiting prescribed Gaussian curvature at all interior vertices and prescribed geodesic curvature along the…

We study the random simple connected cubic planar graph $\mathsf{C}_n$ with an even number $n$ of vertices. We show that the Brownian map arises as Gromov--Hausdorff--Prokhorov scaling limit of $\mathsf{C}_n$ as $n \in 2 \ndN$ tends to…

On a compact Riemannian manifold with boundary, we study the set of conformal metrics of negative constant scalar curvature in the interior and positive constant mean curvature on the boundary. Working in the case of positive Yamabe…

We approximate the uniform measure on an equilateral triangle by a measure supported on $n$ points. We find the optimal sets of points ($n$-means) and corresponding approximation (quantization) error for $n\leq4$, give numerical…

Teramoto et al. defined a new measure called the gap ratio that measures the uniformity of a finite point set sampled from $\cal S$, a bounded subset of $\mathbb{R}^2$. We generalize this definition of measure over all metric spaces by…

We study continuum percolation of overlapping circular discs of two sizes. We propose a phenomenological scaling equation for the increase in the effective size of the larger discs due to the presence of the smaller discs. The critical…

We introduce and study the random non-compact metric space called the Brownian plane, which is obtained as the scaling limit of the uniform infinite planar quadrangulation. Alternatively, the Brownian plane is identified as the…