Scale-free percolation is a spatial random graph model with vertex set $mathbb{Z}^d$. Vertices $x$ and $y$ are connected with probability depending on i.i.d. vertex weights and the Euclidean distance. Depending on the various parameters involved, we
get a rich phase diagram. We study graph distances (in comparison to Euclidean distances). Our main attention is on a regime where graph distances are (poly-)logarithmic in the Euclidean distance. We obtain improved bounds on the logarithmic exponents. In the light tail regime, the correct exponent is identified.
We study the sizes of the Voronoi cells of $k$ uniformly chosen vertices in a random split tree of size $n$. We prove that, for $n$ large, the largest of these $k$ Voronoi cells contains most of the vertices, while the sizes of the remaining ones are
essentially all of order $nexp(-mathrm{const}sqrt{log n})$. This discrepancy persists if we modify the definition of the Voronoi cells by (a) introducing random edge lengths (with suitable moment assumptions), and (b) assigning different influence parameters (called speeds in the paper) to each of the $k$ vertices. Our findings are in contrast to corresponding results on random uniform trees and on the continuum random tree, where it is known that the vector of the relative sizes of the $k$ Voronoi cells is asymptotically uniformly distributed on the $(k-1)$-dimensional simplex.
We demonstrate how sophisticated graph properties, such as small distances and scale-free degree distributions, arise naturally from a reinforcement mechanism on layered graphs. Every node is assigned an a-priori i.i.d. fitness with max-stable distri
bution. The fitness determines the node attractiveness w.r.t. incoming edges as well as the spatial range for outgoing edges. For max-stable fitness distributions, we thus obtain complex spatial network, which we coin extremal linkage network.
We expand the critical point for site percolation on the $d$-dimensional hypercubic lattice in terms of inverse powers of $2d$, and we obtain the first three terms rigorously. This is achieved using the lace expansion.
We investigate a large class of random graphs on the points of a Poisson process in $d$-dimensional space, which combine scale-free degree distributions and long-range effects. Every Poisson point carries an independent random weight and given weight
and position of the points we form an edge between two points independently with a probability depending on the two weights and the distance of the points. In dimensions $din{1,2}$ we completely characterise recurrence vs transience of random walks on the infinite cluster. In $dgeq 3$ we prove transience in all cases except for a regime where we conjecture that scale-free and long-range effects play no role. Our results are particularly interesting for the special case of the age-dependent random connection model recently introduced in [P. Gracar et al., The age-dependent random connection model, Queueing Syst. {bf 93} (2019), no.~3-4, 309--331. MR4032928].
There are various notions of dimension in fractal geometry to characterise (random and non-random) subsets of $mathbb R^d$. In this expository text, we discuss their analogues for infinite subsets of $mathbb Z^d$ and, more generally, for infinite gra
phs. We then apply these notions to critical percolation clusters, where the various dimensions have different values.
We study the random connection model driven by a stationary Poisson process. In the first part of the paper, we derive a lace expansion with remainder term in the continuum and bound the coefficients using a new version of the BK inequality. For our main results, we consider thr
We investigate a model for opinion dynamics, where individuals (modeled by vertices of a graph) hold certain abstract opinions. As time progresses, neighboring individuals interact with each other, and this interaction results in a realignment of opi
nions closer towards each other. This mechanism triggers formation of consensus among the individuals. Our main focus is on strong consensus (i.e. global agreement of all individuals) versus weak consensus (i.e. local agreement among neighbors). By extending a known model to a more general opinion space, which lacks a central opinion acting as a contraction point, we provide an example of an opinion formation process on the one-dimensional lattice with weak consensus but no strong consensus.
We consider the ferromagnetic nearest-neighbor Ising model on regular trees (Bethe lattice), which is well-known to undergo a phase transition in the absence of an external magnetic field. The behavior of the model at critical temperature can be desc
ribed in terms of various critical exponents; one of them is the critical 1-arm exponent $rho$, which characterizes the rate of decay of the (root) magnetization. The crucial quantity we analyze in this work is the thermal expectation of the root spin on a finite subtree, where the expected value is taken with respect to a probability measure related to the corresponding finite-volume Hamiltonian with a fixed boundary condition. The spontaneous magnetization, which is the limit of this thermal expectation in the distance between the root and the boundary (i.e. in the height of the subtree), is known to vanish at criticality. We are interested in a quantitative analysis of the rate of this convergence in terms of the critical 1-arm exponent $rho$. Therefore, we rigorously prove that $langlesigma_0rangle^+_n$, the thermal expectation of the root spin at the critical temperature and in the presence of the positive boundary condition, decays as $langlesigma_0rangle^+_napprox n^{-1/2}$ (in a rather sharp sense), where $n$ is the height of the tree. This establishes the 1-arm critical exponent for the Ising model on regular trees ($rho=1/2$).
The 1-arm exponent $rho$ for the ferromagnetic Ising model on $mathbb{Z}^d$ is the critical exponent that describes how fast the critical 1-spin expectation at the center of the ball of radius $r$ surrounded by plus spins decays in powers of $r$. Sup
pose that the spin-spin coupling $J$ is translation-invariant, $mathbb{Z}^d$-symmetric and finite-range. Using the random-current representation and assuming the anomalous dimension $eta=0$, we show that the optimal mean-field bound $rhole1$ holds for all dimensions $d>4$. This significantly improves a bound previously obtained by a hyperscaling inequality.
