We define a scaling limit of the height function on the domino
tiling model (dimer model) on simply connected regions in $\mathbf{Z}^2$ and
show that it is the “massless free field,” a Gaussian process
with independent coefficients when expanded in the eigenbasis of the
Laplacian.
References
[1] Aizenman, M., Burchard, A., Newman, C. M. and Wilson, D. B. (1999). Scaling limits for minimal and random spanning trees in two dimensions. Statistical physics methods in discrete probability, combinatorics, and theoretical computer science (Princeton, NJ,
1997). Random Structures Algorithm 15 319-367.
[2] Billingsley, P. (1979). Probability and Measure. Wiley, New York.
[3] Burton, R. and Pemantle, R. (1993). Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer-impedances. Ann. Probab. 21 1329-1371.
[4] Itzykson, C. and Drouffe, J.-M. (1989). Th´eorie statistique des champs I. Editions du
[5] Kenyon, R. (2000). Conformal invariance of domino tiling. Ann. Probab. 28 759-795.
[6] Kenyon, R. (2000). Long-range properties of spanning trees. Probabilistic techniques in equilibrium and nonequilibrium statistical physics. J. Math. Phys. 41 1338-1363.
[7] Kenyon, R., Propp, J. and Wilson, D. (2000). Trees and matchings. Electron. J. Combin. 7.
[8] Levy, P. (1965). Processus stochastiques et mouvement brownien. Gauthier-Villars, Paris.
[9] Nienhuis, B. (1984). Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas. J. Statist. Phys. 34 731-761.
[10] Nienhuis, B. (1987). Coulomb gas formulation of two-dimensional phase transitions. In Phase Transitions and Critical Phenomena 11 1-53. Academic Press, London.
[11] Pr¨ahofer, M. and Spohn, H. (1997). An exactly solved model of three-dimensional surface growth in the anisotropic KPZ regime. J. Statist. Phys. 88 999-1012.
[12] Schramm, O. (2000). Scaling limits of loop-erased random walks and uniform spanning trees. Israel. J. Math. 118 221-288.
[13] Simon, B. (1974). The P 2 Euclidean (Quantum) Field Theory. Princeton Univ. Press.
[14] Temperley, H. (1974). Combinatorics: Proceedings of the British Combinatorial Conference. London Math. Soc. Lecture Notes Ser. 13 202-204.
[15] Tesler, G. (2000). Matchings in graphs on non-orientable surfaces. J. Combin. Theory Ser. B 78 198-231.
[16] Thurston, W. P. (1990). Conway's tiling groups. Amer. Math. Monthly 97 757-773.
