Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Critical Ising model and spanning trees partition functions

Béatrice de Tilière

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We prove that the squared partition function of the two-dimensional critical Ising model defined on a finite, isoradial graph $\mathsf{G}=(\mathsf{V},\mathsf{E})$, is equal to $2^{|\mathsf{V}|}$ times the partition function of spanning trees of the graph $\bar{\mathsf{G}}$, where $\bar{\mathsf{G}}$ is the graph $\mathsf{G}$ extended along the boundary; edges of $\mathsf{G}$ are assigned Kenyon’s (Invent. Math. 150 (2) (2002) 409–439) critical weights, and boundary edges of $\bar{\mathsf{G}}$ have specific weights. The proof is an explicit construction, providing a new relation on the level of configurations between two classical, critical models of statistical mechanics.


Nous montrons que le carré de la fonction de partition du modèle d’Ising critique en dimension deux, défini sur un graphe isoradial $\mathsf{G}=(\mathsf{V},\mathsf{E})$ fini, est égale à $2^{|\mathsf{V}|}$ fois la fonction de partition des arbres couvrants du graphe $\bar{\mathsf{G}}$, où le graphe $\bar{\mathsf{G}}$ est le graphe $\mathsf{G}$ prolongé le long du bord; les arêtes de $\mathsf{G}$ sont munies des poids critiques de Kenyon (Invent. Math. 150 (2) (2002) 409–439), et les arêtes du bord de $\bar{\mathsf{G}}$ ont des poids spécifiques. La preuve consiste en une construction explicite, qui donne une nouvelle relation, au niveau des configurations, entre deux modèles classiques de mécanique statistique au point critique.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 3 (2016), 1382-1405.

Received: 20 January 2014
Revised: 23 February 2015
Accepted: 8 April 2015
First available in Project Euclid: 28 July 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82B27: Critical phenomena 05A19: Combinatorial identities, bijective combinatorics

Critical two-dimensional Ising model Critical spanning trees Isoradial graphs Partition functions


de Tilière, Béatrice. Critical Ising model and spanning trees partition functions. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 3, 1382--1405. doi:10.1214/15-AIHP680.

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