Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

The Green’s function on the double cover of the grid and application to the uniform spanning tree trunk

Richard W. Kenyon and David B. Wilson

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We compute the Green’s function on the double cover of $\mathbb{Z}^{2}$, branched over a vertex or a face. We use this result to compute the local statistics of the “trunk” of the uniform spanning tree on the square lattice, i.e., the limiting probabilities of cylinder events conditional on the path connecting far away points passing through a specified edge. We also show how to compute the local statistics of large-scale triple points of the uniform spanning tree, where the trunk branches. The method reduces the problem to a dimer system with isolated monomers, and we compute the inverse Kasteleyn matrix using the Green’s function on the double cover of the square lattice. For the trunk, the probabilities of cylinder events are in $\mathbb{Q}[\sqrt{2}]$, while for the triple points the probabilities are in $\mathbb{Q}[1/\pi ]$.


Nous calculons la fonction de Green sur le revêtement à deux feuillets de $\mathbb{Z}^{2}$, ramifié au-dessus d’un sommet ou d’une face. Nous utilisons ce résultat pour calculer les statistiques locales du « tronc » de l’arbre couvrant minimal sur le réseau carré, c’est-à-dire les probabilités limites des événements cylindriques conditionnées à ce que le chemin connectant deux sommets éloignés passe par une arête donnée. Nous montrons également comment calculer les statistiques locales des points triples à grande échelle de l’arbre couvrant minimal, où le tronc se sépare. La méthode consiste à ramener le problème à un système de dimères avec des monomères isolés, et nous calculons l’inverse de la matrice de Kasteleyn à l’aide de la fonction de Green sur le revêtement deux feuillets du réseau carré. Pour le tronc, les probabilités des événements cylindriques sont dans $\mathbb{Q}[\sqrt{2}]$, tandis que pour les points triples, les probabilités sont dans $\mathbb{Q}[1/\pi]$.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 56, Number 3 (2020), 1841-1868.

Received: 11 September 2018
Revised: 3 July 2019
Accepted: 16 August 2019
First available in Project Euclid: 26 June 2020

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]

Loop-erased random walk Laplacian Green’s function


Kenyon, Richard W.; Wilson, David B. The Green’s function on the double cover of the grid and application to the uniform spanning tree trunk. Ann. Inst. H. Poincaré Probab. Statist. 56 (2020), no. 3, 1841--1868. doi:10.1214/19-AIHP1019.

Export citation


  • [1] C. Beneš, G. F. Lawler and F. Viklund. Scaling limit of the loop-erased random walk Green’s function. Probab. Theory Related Fields 166 (1–2) (2016) 271–319. Available at arXiv:1402.7345.
  • [2] I. Benjamini, R. Lyons, Y. Peres and O. Schramm. Uniform spanning forests. Ann. Probab. 29 (1) (2001) 1–65.
  • [3] M. Bousquet-Mélou. Walks on the slit plane: Other approaches. Adv. in Appl. Math. 27 (2–3) (2001) 243–288. Special issue in honor of Dominique Foata’s 65th birthday. Available at math/0104111.
  • [4] M. Bousquet-Mélou and G. Schaeffer. Walks on the slit plane. Probab. Theory Related Fields 124 (3) (2002) 305–344. Available at math/0012230.
  • [5] T. Budd. Winding of simple walks on the square lattice, 2017. Available at arXiv:1709.04042.
  • [6] R. Burton and R. Pemantle. Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer-impedances. Ann. Probab. 21 (3) (1993) 1329–1371.
  • [7] D. Chelkak, D. Cimasoni and A. Kassel. Revisiting the combinatorics of the 2D Ising model. Ann. Inst. Henri Poincaré D 4 (3) (2017) 309–385. Available at arXiv:1507.08242.
  • [8] P. G. Doyle and J. L. Snell. Random Walks and Electric Networks. Carus Mathematical Monographs 22. Mathematical Association of America, Washington, DC, 1984.
  • [9] M. E. Fisher and J. Stephenson. Statistical mechanics of dimers on a plane lattice. II. Dimer correlations and monomers. Phys. Rev. 2 (132) (1963) 1411–1431.
  • [10] R. Gheissari, C. Hongler and S. C. Park. Ising model: Local spin correlations and conformal invariance. Comm. Math. Phys. 367 (3) (2019) 771–833.
  • [11] R. Kenyon. Local statistics of lattice dimers. Ann. Inst. Henri Poincaré Probab. Stat. 33 (5) (1997) 591–618. Available at math/0105054.
  • [12] R. Kenyon. The asymptotic determinant of the discrete Laplacian. Acta Math. 185 (2) (2000) 239–286. Available at math-ph/0011042.
  • [13] R. Kenyon. Conformal invariance of domino tiling. Ann. Probab. 28 (2) (2000) 759–795. Available at math-ph/9910002.
  • [14] R. Kenyon. Long-range properties of spanning trees. J. Math. Phys. 41 (3) (2000) 1338–1363.
  • [15] R. Kenyon. Lectures on dimers. In Statistical Mechanics 191–230. IAS/Park City Math. Ser. 16. Am. Math. Soc., Providence, 2009.
  • [16] R. W. Kenyon, J. G. Propp and D. B. Wilson. Trees and matchings. Electron. J. Combin. 7 (2000) 25.
  • [17] R. W. Kenyon and D. B. Wilson. Spanning trees of graphs on surfaces and the intensity of loop-erased random walk on planar graphs. J. Amer. Math. Soc. 28 (4) (2015) 985–1030. Available at arXiv:1107.3377.
  • [18] G. F. Lawler. The probability that planar loop-erased random walk uses a given edge. Electron. Commun. Probab. 19 (51) (2014) 13. Available at arXiv:1301.5331.
  • [19] G. F. Lawler. The infinite two-sided loop-erased random walk, 2018. Available at arXiv:1802.06667.
  • [20] G. F. Lawler and V. Limic. Random Walk: A Modern Introduction. Cambridge Studies in Advanced Mathematics 123. Cambridge University Press, Cambridge, 2010.
  • [21] R. Lyons and J. E. Steif. Stationary determinantal processes: Phase multiplicity, Bernoullicity, entropy, and domination. Duke Math. J. 120 (3) (2003) 515–575.
  • [22] W. H. McCrea and F. J. W. Whipple. Random paths in two and three dimensions. Proc. R. Soc. Edinb. 60 (1940) 281–298.
  • [23] R. Pemantle. Choosing a spanning tree for the integer lattice uniformly. Ann. Probab. 19 (4) (1991) 1559–1574.
  • [24] O. Schramm. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 (2000) 221–288.
  • [25] F. Spitzer. Principles of Random Walk, 2nd edition. Graduate Texts in Mathematics 34. Springer-Verlag, Berlin, 1976.
  • [26] A. Stöhr. Über einige lineare partielle Differenzengleichungen mit konstanten Koeffizienten. III. Zweites Beispiel: Der Operator $\nabla \Phi (y_{1},y_{2})=\Phi (y_{1}{+}1,y_{2})+\Phi (y_{1}{-}1,y_{2})+\Phi (y_{1},y_{2}{+}1)+\Phi (y_{1},y_{2}{-}1)-4\Phi (y_{1},y_{2})$. Math. Nachr. 3 (1950) 330–357.
  • [27] H. N. V. Temperley In Combinatorics: Proceedings of the British Combinatorial Conference 1973 202–204. London Mathematical Society Lecture Notes Series 13, 1974.