Related papers: The radial spanning tree in hyperbolic space
Bootstrap percolation is a type of cellular automaton which has been used to model various physical phenomena, such as ferromagnetism. For each natural number $r$, the $r$-neighbour bootstrap process is an update rule for vertices of a…
We consider a stationary Poisson process of $k$-planes in the $d$-dimensional hyperbolic space $\mathbb H^d$ of constant curvature $-1$, with $d \ge 4$ and $1 \le k \le d-1$. It is known that, after centring and normalization, the total…
We consider random graphs with uniformly bounded edges on a Poisson point process conditioned to contain the origin. In particular we focus on the random connection model, the Boolean model and Miller-Abrahams random resistor network with…
We provide sufficient conditions for a regular graph $G$ of growing degree $d$, guaranteeing a phase transition in its random subgraph $G_p$ similar to that of $G(n,p)$ when $p\cdot d\approx 1$. These conditions capture several well-studied…
In this work, we study the color discrepancy of spanning trees in random graphs. We show that for the Erd\H{o}s-R\'enyi random graph $G(n,p)$ with $p$ above the connectivity threshold, the following holds with high probability: in every…
We consider the problem of enumerating spanning trees on lattices. Closed-form expressions are obtained for the spanning tree generating function for a hypercubic lattice of size N_1 x N_2 x...x N_d in d dimensions under free, periodic, and…
We show that every connected graph can be approximated by a normal tree, up to some arbitrarily small error phrased in terms of neighbourhoods around its ends. The existence of such approximate normal trees has consequences of both…
The Euclidean Directed Spanning Forest is a random forest in $\mathbb{R}^d$ introduced by Baccelli and Bordenave in 2007 and we introduce and study here the analogous tree in the hyperbolic space. The topological properties of the Euclidean…
Uniform spanning trees are a statistical model obtained by taking the set of all spanning trees on a given graph (such as a portion of a cubic lattice in d dimensions), with equal probability for each distinct tree. Some properties of such…
The intricate relations between elements in natural and human-made systems sustain the complex processes that shape our world, forming multiscale networks of interactions. These networks can be represented as graphs composed of nodes…
If $X$ is a geodesic metric space and $x_1,x_2,x_3\in X$, a geodesic triangle $T=\{x_1,x_2,x_3\}$ is the union of the three geodesics $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$ in $X$. The space $X$ is $\delta$-hyperbolic (in the Gromov sense)…
For a connected network on Poisson points in the plane, consider the route-length $D(r,\theta) $ between a point near the origin and a point near polar coordinates $(r,\theta)$, and suppose $E D(r,\theta) = O(r)$ as $r \to \infty$. By…
Undirected hyperbolic graph models have been extensively used as models of scale-free small-world networks with high clustering coefficient. Here we presented a simple directed hyperbolic model, where nodes randomly distributed on a…
We study level-set percolation of the Gaussian free field on the infinite $d$-regular tree for fixed $d\geq 3$. Denoting by $h_\star$ the critical value, we obtain the following results: for $h>h_\star$ we derive estimates on conditional…
We show that many graphs naturally associated to a connected, compact, orientable surface are hierarchically hyperbolic spaces in the sense of Behrstock, Hagen and Sisto. They also automatically have the coarse median property defined by…
We describe the asymptotic behaviour of large degrees in random hyperbolic graphs, for all values of the curvature parameter $ \alpha$. We prove that, with high probability, the node degrees satisfy the following ordering property: the…
In this paper we obtain precise asymptotics for certain families of graphs, namely circulant graphs and degenerating discrete tori. The asymptotics contain interesting constants from number theory among which some can be interpreted as…
A random geometric graph $G(\mathcal{X}_n, r_n)$ is formed by taking a binomial process $\mathcal{X}_n$ as the set of vertices and joining any two distinct points with an edge if they lie within distance $r_n$ of each other. We investigate…
In this paper, we explore how geometric structures can be grown exponentially fast. The studied processes start from an initial shape and apply a sequence of centralized growth operations to grow other shapes. We focus on the case where the…
In this paper, we compare the solutions of Dyson Brownian motion with general $\beta$ and potential $V$ and the associated McKean-Vlasov equation near the edge. Under suitable conditions on the initial data and potential $V$, we obtain the…