The primal versus the dual ising model
Pages 53 - 60
Abstract
We represent the Ising model of statistical physics by normal factor graphs in the primal and in the dual domains. By analogy with Kirchhoff's voltage and current laws, we show that in the primal normal factor graphs, the dependency among the variables is along the cycles, whereas in the dual normal factor graphs, the dependency is on the cutsets. In the primal (resp. dual) domain, dependent variables can be computed via their fundamental cycles (resp. fundamental cutsets). Using Onsager's closed form solution, we illustrate the opposite behavior of the uniform sampling estimator for estimating the partition function in the primal and in the dual of the homogeneous Ising model on a two-dimensional torus.
References
[1]
G.D. Forney, Jr., “Codes on graphs: normal realization,” IEEE Trans. on Information Theory, vol. 47, pp. 520–548, Feb. 2001.
[2]
B.A. Cipra, “An introduction to the Ising model,” American Math. Monthly, vol. 94, pp. 937–959, Dec. 1987.
[3]
R.J. Baxter, Exactly Solved Models in Statistical Mechanics. Dover Publications, 2007.
[4]
L. Onsager, “Crystal statistics. I. A two-dimensional model with an order-disorder transition,” Phys. Rev., vol. 65, pp. 117–149, Feb. 1944.
[5]
B. Bollobás, Modern Graph Theory. Spriger, 1998.
[6]
C. Godsil and G. Royle, Algebraic Graph Theory. Spriger, 2001.
[7]
A. Al-Bashabsheh and P. Vontobel, “The Ising model: Kramers-Wannier duality and normal factor graphs,” Proc. 2015 IEEE Int. Symp. on Information Theory, Hong Kong, June 14–19, 2015, pp. 2266–2270.
[8]
A. Al-Bashabsheh and P. Vontobel, “A factor-graph approach to algebraic topology, with applications to Kramers-Wannier duality,” arXiv:, 2017.
[9]
G.D. Forney, Jr., “Graphical models for elementary algebraic topology, with applications to statistical physics and codes on graphs,” arXiv:, 2017.
[10]
F.Y. Wu and Y.K. Wang, “Duality transformation in a many-component spin model,” Journal of Math. Phys., vol. 17, pp. 439–440, March 1976.
[11]
I. Csiszár and P.C. Shields, Information Theory and Statistics: A Tutorial. Foundations and Trends in Communications and Information Theory, vol. 1, pp. 417–528, Dec. 2004.
[12]
A. Al-Bashabsheh and Y. Mao, “Normal factor graphs and holographic transformations,” IEEE Trans. on Information Theory, vol. 57, pp. 752–763, Feb. 2011.
[13]
M. Molkaraie and H.-A. Loeliger, “Partition function of the Ising model via factor graph duality,” Proc. 2013 IEEE Int. Symp. on Information Theory, Istanbul, Turkey, July 7–12, 2013, pp. 2304–2308.
[14]
A. Al-Bashabsheh and Y. Mao, “On stochastic estimation of the partition function,” Proc. 2014 IEEE Int. Symp. on Information Theory, Honolulu, USA, June 29–July 4, 2014, pp. 1504–1508.
[15]
M. Molkaraie, “An importance sampling scheme for models in a strong external field,” Proc. 2015 IEEE Int. Symp. on Information Theory, Hong Kong, June 14–19, 2015, pp. 1179–1183.
[16]
M. Molkaraie, “An importance sampling algorithm for the Ising model with strong couplings,” Proc. 2016 Int. Zurich Seminar on Communications, March 2–4, 2016, pp. 180–184.
[17]
D.W. Kammler, A First Course in Fourier Analysis. Cambridge University Press, 2007.
Index Terms
- The primal versus the dual ising model
Index terms have been assigned to the content through auto-classification.
Recommendations
Dual–primal algorithm for linear optimization
The purpose of this paper is to present a new approach for solving linear programming, which has some interesting theoretical properties. In each step of the iteration, we trace a direction completely different from primal simplex method, dual simplex ...
Primal--dual methods for linear programming
Many interior-point methods for linear programming are based on the properties of the logarithmic barrier function. After a preliminary discussion of the convergence of the (primal) projected Newton barrier method, three types of barrier method are ...
Monotonicity of Primal and Dual Objective Values in Primal-dual Interior-point Algorithms
Monotonicity of primal and dual objective values in the framework of primal-dual interior-point methods is studied. The primal-dual affine-scaling algorithm is monotone in both objectives. A condition is derived under which a primal-dual interior-point ...
Comments
Information & Contributors
Information
Published In
Oct 2017
1304 pages
Copyright © 2017.
Publisher
IEEE Press
Publication History
Published: 03 October 2017
Qualifiers
- Research-article
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 0Total Downloads
- Downloads (Last 12 months)0
- Downloads (Last 6 weeks)0
Reflects downloads up to 27 Jul 2024
Other Metrics
Citations
View Options
View options
Get Access
Login options
Check if you have access through your login credentials or your institution to get full access on this article.
Sign in