44 citations · 181 across the 27 of their papers we have counts for
7 papers · 1 filter
Fractal dimension of discrete sets and percolation
Markus Heydenreich
There are various notions of dimension in fractal geometry to characterise (random and non-random) subsets of . In this expository text, we discuss their analogues for…
Expansion for the critical point of site percolation: the first three terms
Markus Heydenreich, Kilian Matzke
We expand the critical point for site percolation on the -dimensional hypercubic lattice in terms of inverse powers of , and we obtain the first three terms rigorously. This…
Recurrence versus Transience for Weight-Dependent Random Connection Models
Peter Gracar, Markus Heydenreich, Christian Mönch +1
We investigate random graphs on the points of a Poisson process in -dimensional space, which combine scale-free degree distributions and long-range effects. Every Poisson point…
Critical site percolation in high dimension
Markus Heydenreich, Kilian Matzke
We use the lace expansion to prove an infra-red bound for site percolation on the hypercubic lattice in high dimension. This implies the triangle condition and allows us to derive…
Lace Expansion and Mean-Field Behavior for the Random Connection Model
Markus Heydenreich, Remco van der Hofstad, Günter Last +1
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 bou…
A spatial small-world graph arising from activity-based reinforcement
Markus Heydenreich, Christian Hirsch
In the classical preferential attachment model, links form instantly to newly arriving nodes and do not change over time. We propose a hierarchical random graph model in a spatial…