# 双体模型

8x8平方骨牌密鋪

## 介绍

${\displaystyle Z={\sqrt {|\det K|}}=\prod _{j=1}^{\lceil {\frac {m}{2}}\rceil }\prod _{k=1}^{\lceil {\frac {n}{2}}\rceil }\left(4\cos ^{2}{\frac {\pi j}{m+1}}+4\cos ^{2}{\frac {\pi k}{n+1}}\right).}$

K是G的邻接矩阵。 Z也是统计力学的配分函数[7]

• ${\displaystyle 2\times n}$ 格子${\displaystyle Z_{n}}$斐波那契数列 [8]
• ${\displaystyle m=n=2k=0,2,4,\ldots }$，可以使用普法夫值计算Z [9]

${\displaystyle \lim _{m,n\to \infty }Z(m,n)/(mn)=C/\pi }$

## 阿兹特克钻石与北极圈现象

Z也依赖格子的边界（参看阿兹特克钻石英语Aztec diamond）。

## 高度函数

${\displaystyle s(v)=\pm 1}$

s是自旋（参看易辛模型）、v是顶点。那么可以定义一个1-形式

${\displaystyle ds(e=uv)=s(u)-s(v)=\pm 2}$

${\displaystyle \delta (e)=0,1}$

${\displaystyle dh(e)=h(u)-h(v)=(1+\delta (e))ds(e)=\pm 3}$

NxN平方格子的高度函数在中间逼近O(N)。但是阿兹特克钻石的高度函数逼近h的平均值。[7]的确，CKP定理[7]说h最小化一个（或热力学自由能）的泛函变分法）：

${\displaystyle F=\log Z}$

## 共形场论

### 高斯自由场

${\displaystyle Kf=f(w+1)-f(w-1)+if(w+i)-if(w-i)=0}$

f是“反全纯函数”。再说 f 是调和函数和谐函数）。这是因为${\displaystyle KK^{*}=L}$调和矩阵（harmonic matrix）。[7]

### 传播子

${\displaystyle D(x,y)=\langle h(x)h(y)\rangle }$

${\displaystyle D(u,v)=-{\frac {1}{2\pi }}\log(v-u)}$

${\displaystyle d_{u}D(u,v)={\frac {1}{2\pi }}({\frac {du}{v-u}}-{\frac {d{\bar {u}}}{v-{\bar {u}}}})}$

${\displaystyle \langle h(x_{1})\ldots h(x_{2k})\rangle \propto \sum _{\text{对 }}D(x_{i_{1}}x_{i_{2}})\ldots D(x_{i_{2k-1}}x_{i_{2k}})}$

## 参考文献

1. Richard Kenyon and Andrei Okounkov. What is a dimer? (PDF).
2. ^ Baake, Michael.; Moody, R. V., 1941-. Directions in mathematical quasicrystals. Providence, R.I.: American Mathematical Society https://www.worldcat.org/oclc/45248226. 2000. ISBN 0-8218-2629-8. OCLC 45248226. 缺少或|title=为空 (帮助)
3. Richard Kenyon. The planar dimer model with boundary: a survey (PDF) （英语）.
4. Introduction to Random Tilings. faculty.uml.edu. [2020-02-14].
5. ^ Temperley, H. N. V.; Fisher, Michael E. Dimer problem in statistical mechanics-an exact result. Philosophical Magazine. 1961-08, 6 (68): 1061–1063. ISSN 0031-8086. doi:10.1080/14786436108243366 （英语）.
6. ^ Kasteleyn, P.W. The statistics of dimers on a lattice. Physica. 1961-12, 27 (12): 1209–1225. doi:10.1016/0031-8914(61)90063-5 （英语）.
7. Kenyon, Richard. An introduction to the dimer model. 2003-10-20 （英语）.
8. ^ A000045 - OEIS. oeis.org. [2020-02-13].
9. ^ A004003 - OEIS. oeis.org. [2020-02-13].

## 阅读

• R. Baxter, Exactly solved models in statistical mechanics. Academic Press.
• Olivier Bodini, Matthieu Latapy. Generalized Tilings with Height Functions // Morfismos. — 2003. — Т. 7, вып. 1. — С. 47–68. — ISSN 1870-6525.
• F. Faase. On the number of specific spanning subgraphs of the graphs G X P_n // Ars Combin.. — 1998. — Т. 49. — С. 129–154.
• J. L. Hock, R. B. McQuistan. A note on the occupational degeneracy for dimers on a saturated two-dimenisonal lattice space // Discrete Appl. Math.. — 1984. — Т. 8. — С. 101–104. — DOI:10.1016/0166-218X(84)90083-0.
• P. W. Kasteleyn. The statistics of dimers on a lattice. I. The number of dimer arrangements on a quadratic lattice // Physica. — 1961. — Т. 27, вып. 12. — С. 1209–1225. — DOI:10.1016/0031-8914(61)90063-5. —Bibcode: 1961Phy....27.1209K..
• Richard Kenyon. Directions in mathematical quasicrystals / Michael Baake, Robert V. Moody. — Providence, RI: American Mathematical Society, 2000. — Т. 13. — С. 307–328. — ISBN 0-8218-2629-8.
• Richard Kenyon, Andrei Okounkov. What is … a dimer? // Notices of the American Mathematical Society. — 2005. — Т. 52, вып. 3. — P. 342–343. — ISSN 0002-9920..
• David Klarner, Jordan Pollack. Domino tilings of rectangles with fixed width // Discrete Mathematics. — 1980. — Т. 32, вып. 1. — DOI:10.1016/0012-365X(80)90098-9..
• Richard J. Paving rectangular regions with rectangular tiles: tatami and non-tatami tilings. — 2013.
• Lambda-determinants and domino-tilings // Advances in Applied Mathematics. — 2005. — Т. 34, вып. 4. — С. 871–879. — DOI:10.1016/j.aam.2004.06.005. — arXiv:math.CO/0406301..
• Frank Ruskey, Jennifer Woodcock. Counting fixed-height Tatami tilings. — 2009. — Т. 16, вып. 1. — С. R126.
• James A. Sellers. Domino tilings and products of Fibonacci and Pell numbers // Journal of Integer Sequences. — 2002. — Т. 5, вып. Article 02.1.2..
• Richard P. Stanley. On dimer coverings of rectangles of fixed width // Discrete Appl. Math. — 1985. — Т. 12. — С. 81–87. — DOI:10.1016/0166-218x(85)90042-3.
• W. P. Thurston.威廉·瑟斯顿）Conway's tiling groups. — American Mathematical Monthly. — Mathematical Association of America, 1990. — Т. 97. — С. 757–773. — DOI:10.2307/2324578..
• David Wells. The Penguin Dictionary of Curious and Interesting Numbers. — London: Penguin, 1997. — С. 182. — ISBN 0-14-026149-4..
• H. N. V. Temperley, Michael E. Fisher. Dimer problem in statistical mechanics-an exact result // Philosophical Magazine. — 1961. — Т. 6, вып. 68. — С. 1061–1063. — DOI:10.1080/14786436108243366.
• Erickson, Alejandro; Ruskey, Frank (2013), "Domino tatami covering is NP-complete", Combinatorial algorithms, Lecture Notes in Comput. Sci., 8288, Springer, Heidelberg, pp. 140–149, arXiv:1305.6669, doi:10.1007/978-3-642-45278-9_13, MR 3162068