Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Random two-component spanning forests

Adrien Kassel, Richard Kenyon, and Wei Wu

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We study random two-component spanning forests ($2$SF) of finite graphs, giving formulas for the first and second moments of the sizes of the components, vertex-inclusion probabilities for one or two vertices, and the probability that an edge separates the components. We compute the limit of these quantities when the graph tends to an infinite periodic graph in $\mathbb{R}^{d}$.


Nous étudions la mesure uniforme sur les forêts couvrantes à deux composantes connexes d’un graphe fini et donnons des formules pour les deux premiers moments de la taille des composantes, les probabilités d’inclusion d’un ou deux sommets dans la même composante, et la probabilité qu’une arête sépare les composantes. Nous calculons la limite des ces quantités lorsque l’on considère une suite de graphes finis qui tend vers un graphe infini périodique dans $\mathbb{R}^{d}$.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 51, Number 4 (2015), 1457-1464.

Received: 26 September 2013
Revised: 21 April 2014
Accepted: 27 April 2014
First available in Project Euclid: 21 October 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 05C81: Random walks on graphs

Two-component spanning forests Mean resistance Torsional rigidity


Kassel, Adrien; Kenyon, Richard; Wu, Wei. Random two-component spanning forests. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 4, 1457--1464. doi:10.1214/14-AIHP625.

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