Geodesic
Asymptotic properties of random unlabelled block-weighted graphs
The electronic journal of combinatorics, Tome 28 (2021) no. 4
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We study the asymptotic shape of random unlabelled graphs subject to certain subcriticality conditions. The graphs are sampled with probability proportional to a product of Boltzmann weights assigned to their $2$-connected components. As their number of vertices tends to infinity, we show that they admit the Brownian tree as Gromov–Hausdorff–Prokhorov scaling limit, and converge in a strengthened Benjamini–Schramm sense toward an infinite random graph. We also consider models of random graphs that are allowed to be disconnected. Here a giant connected component emerges and the small fragments converge without any rescaling towards a finite random limit graph. Our main application of these general results treats subcritical classes of unlabelled graphs. We study the special case of unlabelled outerplanar graphs in depth and calculate its scaling constant.
DOI : 10.37236/9923
Classification : 60C05, 05C80
Mots-clés : Boltzmann weights 
@article{10_37236_9923,
     author = {Benedikt Stufler},
     title = {Asymptotic properties of random unlabelled block-weighted graphs},
     journal = {The electronic journal of combinatorics},
     year = {2021},
     volume = {28},
     number = {4},
     doi = {10.37236/9923},
     zbl = {1494.60014},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/9923/}
}
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Benedikt Stufler. Asymptotic properties of random unlabelled block-weighted graphs. The electronic journal of combinatorics, Tome 28 (2021) no. 4. doi: 10.37236/9923

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