time: Mondays 3-6 PM
place: Boelter 5420

Brief description:

The prime goal of this seminar is to get acquainted with the details of (predominantly Kenyon's) work on counting, and determining various statistical properties, of domino tilings -- or, more generally, perfect matchings. Below is the list of relevant references. The history is roughly as follows: In 1961 Kasteleyn proved a theorem concerning the number of perfect matchings in planar graphs. In the 1990s, Burton and Permantle proved results that built upon (presumably known) connection with spanning trees while Kenyon realized that Kasteleyn's technique permits the derivation of various local statistics of uniformly-random perfect matchings. Later he extended this to the proof of conformal invariance for the height function of domino tilings. Finally, the last 5 years have seen classifications of domino measures according to the boundary conditions (the work of Cohn, Kenyon, Okounkov, Propp and Sheffield), and applications to the interface in the 3D Ising model (Cerf-Kenyon). It can be expected that in the next couple of years we will see further links to conformal invariance and statistical mechanics at the critical point.

Order of presentations: (click on name for extended abstract)

1. October 10, 2005: Jason Asher
2. October 17, 2005: Alex Smith
3. October 24, 2005: Vira Kalinichenko
4. October 31, 2005: Helen Lei
5. November 7, 2005: Paul Jones
6. November 14, 2005: Jon Handy
7. November 21, 2005: Nick Crawford
8. November 28, 2005: Jason Lenderman
9. December 5, 2005: Tim Prescott

Papers to read:

1. P. Kasteleyn, The statistics of dimers on a lattice. I. The number of dimer arrangements on a quadratic lattice, Physica 27 (1961) 1209--1225. download [Jason Asher]


2. R. Kenyon, Local statistics of lattice dimers, Ann. Inst. H. Poincare Probab. Statist. 33 (1997), no. 5, 591--618. download [Jason Asher]


3. R. Burton and R. Pemantle, Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer-impedances, Ann. Probab. 21 1329--1371. download [Alex Smith]


4. R. J. Duffin, Basic properties of discrete analytic functions, Duke Math. J. 23 (1956), no. 2, 335-363. download [Vira Kalinichenko]


5. R. Kenyon, Conformal invariance of domino tiling, Ann. Probab. 28 (2000), no. 2, 759--795. download [Vira Kalinichenko/Helen Lei]


6. R. Kenyon, Dominos and the Gaussian free field, Ann. Probab. 29 (2001), no. 3, 1128--1137. download [Helen Lei]


7. R. Kenyon, The asymptotic determinant of the discrete Laplacian, Acta Math. 185 (2000), no. 2, 239--286. pdf file [Paul Jones]


8. R.W. Kenyon, J. Propp and D. Wilson, Trees and matchings, Electron. J. Combin. 7 (2000), Research Paper 25, 34 pp. (electronic). download [Jason Lenderman]


9. H. Cohn, R. Kenyon and J. Propp, A variational principle for domino tilings, J. Amer. Math. Soc. 14 (2001), no. 2, 297--346 (electronic). download [Tim Prescott]


10. R. Cerf and R. Kenyon, The low-temperature expansion of the Wulff crystal in the 3D Ising model, Commun. Math. Phys. 222 (2001), no. 1, 147--179. download [Nick Crawford]


11. R. Kenyon, The Laplacian and Dirac operators on critical planar graphs Invent. Math. 150 (2002), no. 2, 409-439. download [Jon Handy]


12. R. Kenyon and S. Sheffield, Dimers, tilings and trees, J. Combin. Theory Ser. B 92 (2004), no. 2, 295--317. download [Jason Lenderman]


13. R. Kenyon, A. Okounkov and S. Sheffield, Dimers and Amoebae, Ann. Math. (to appear) download [Tim Prescott]