Lace expansion and mean-field behavior for the random connection model
- 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 three versions of the connection function φ: a finite-variance version (including the Boolean model), a spread-out version, and a long-range version. For sufficiently large dimension (resp., spread-out parameter and d>6), we then prove the convergence of the lace expansion, derive the triangle condition, and establish an infra-red bound. From this, mean-field behavior of the model can be deduced. As an example, we show that the critical exponent γ takes its mean-field value γ=1 and that the percolation function is continuous.
| Author: | Markus HeydenreichORCiDGND, Remco van der Hofstad, Günter Last, Kilian Matzke |
|---|---|
| Frontdoor URL | https://opus.bibliothek.uni-augsburg.de/opus4/103761 |
| Parent Title (English): | arxiv |
| Publisher: | arXiv |
| Type: | Preprint |
| Language: | English |
| Year of first Publication: | 2022 |
| Release Date: | 2023/04/21 |
| Issue: | arXiv:1908.11356 |
| DOI: | https://doi.org/10.48550/arXiv.1908.11356 |
| Institutes: | Mathematisch-Naturwissenschaftlich-Technische Fakultät |
| Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik | |
| Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik / Lehrstuhl für Stochastik und ihre Anwendungen | |
| Dewey Decimal Classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
| Latest Publications (not yet published in print): | Aktuelle Publikationen (noch nicht gedruckt erschienen) |
