Invariant Schreier decorations of unimodular random networks

Tóth, László Márton

Annales Henri Lebesgue, Tome 4 (2021), pp. 1705-1726.

Nous démontrons que tout graphe aléatoire unimodulaire $2d$-régulier admet une décoration de Schreier aléatoire invariante. De manière équivalent, c’est le graphe de Schreier des classes d’un sous-groupe invariant aléatoire du groupe libre $F_d$. Comme corollaire, nous obtenons que tout graphage $2d$-régulier est localement l’image isomorphe d’un graphage provenant d’une action de $F_d$ préservant une mesure de probabilité.

Les ingrédients-clé de l’énoncé analogue pour les graphes finis ne se généralisent pas tels quels au contexte mesurable. Nous trouvons une manière plus subtile d’adapter ces ingrédients, et nous obtenons en chemin des théorèmes de coloriages mesurables de graphages.

We prove that every $2d$-regular unimodular random network carries an invariant random Schreier decoration. Equivalently, it is the Schreier coset graph of an invariant random subgroup of the free group $F_d$. As a corollary we get that every $2d$-regular graphing is the local isomorphic image of a graphing coming from a p.m.p. action of $F_d$.

The key ingredients of the analogous statement for finite graphs do not generalize verbatim to the measurable setting. We find a more subtle way of adapting these ingredients and prove measurable coloring theorems for graphings along the way.

Classification : 37A50, 20E05, 05C15
Mots clés : Schreier graph, graph limits, unimodular random graph, Invariant Random Subgroup

[AGV14] Abért, Miklós; Glasner, Yair; Virág, Bálint Kesten’s theorem for invariant random subgroups, Duke Math. J., Volume 163 (2014) no. 3, pp. 465-488 | MR | Zbl

[AL07] Aldous, David J.; Lyons, Russell Processes on unimodular random networks, Electron. J. Probab., Volume 12 (2007), pp. 1454-1508 | MR | Zbl

[Bow12] Bowen, Lewis Invariant random subgroups of the free group (2012) (https://arxiv.org/abs/1204.5939v1)

[BS01] Benjamini, Itai; Schramm, Oded Recurrence of Distributional Limits of Finite Planar Graphs, Electron. J. Probab., Volume 6 (2001), 23, p. 13 pp. | DOI | MR | Zbl

[BT17] Biringer, Ian; Tamuz, Omer Unimodularity of invariant random subgroups, Trans. Am. Math. Soc., Volume 369 (2017) no. 6, pp. 4043-4061 | DOI | MR | Zbl

[Can13] Cannizzo, Jan On invariant Schreier structures (2013) (https://arxiv.org/abs/1309.5163v1)

[CK13] Conley, Clinton T.; Kechris, Alexander S. Measurable chromatic and independence numbers for ergodic graphs and group actions, Groups Geom. Dyn., Volume 7 (2013) no. 1, pp. 127-180 | DOI | MR | Zbl

[CL17] Csóka, Endre; Lippner, Gabor Invariant random perfect matchings in Cayley graphs, Groups Geom. Dyn., Volume 11 (2017) no. 1, pp. 211-244 | DOI | MR | Zbl

[CLP16] Csóka, Endre; Lippner, Gabor; Pikhurko, Oleg Kőnig’s line coloring and Vizing’s theorems for graphings, Forum Math. Sigma, Volume 4 (2016), e27 | Zbl

[Ele07] Elek, Gábor On limits of finite graphs, Combinatorica, Volume 27 (2007) no. 4, pp. 503-507 | DOI | MR | Zbl

[HLS14] Hatami, Hamed; Lovász, László; Szegedy, Balázs Limits of locally–globally convergent graph sequences, Geom. Funct. Anal., Volume 24 (2014) no. 1, pp. 269-296 | DOI | MR | Zbl

[KM15] Kechris, Alexander S.; Marks, Andrew S. Descriptive graph combinatorics, 2015 (preprint available at http://www.math.caltech.edu/~kechris/papers/combinatorics20book.pdf)

[KST99] Kechris, Alexander S.; Solecki, Slawomir; Todorcevic, Stevo Borel chromatic numbers, Adv. Math., Volume 141 (1999) no. 1, pp. 1-44 | DOI | MR | Zbl

[Lac88] Laczkovich, Miklós Closed sets without measurable matching, Proc. Am. Math. Soc., Volume 103 (1988) no. 3, pp. 894-896 | DOI | MR | Zbl

[Lov12] Lovász, László Large networks and graph limits, Colloquium Publications, 60, American Mathematical Society, 2012 | Zbl