# Ising model on groups

Can anything interesting be deduced about the properties of a group from the behavior of the Ising model on its Cayley graph? (i.e. existence and character of phase transitions, critical behavior) I'm not sure, though, if one should expect any general results (link between large-scale geometry and universality class, maybe?), since even very simple geometry of the group (say, $\mathbb{Z^2}$) can give highly nontrivial statistical properties.

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I suppose that depends on the particular cayley graph you use. I'm betting that for finite presentations the Ising model should care about word-hyperbolicity; and also that if your group is virtually free abelian then you should get something essentially equivalent to the standard Ising model; but I don't know enough, so I'm not answering. –  some guy on the street Jan 16 '11 at 19:16
Whether the "coarse" properties of the Ising model depend on a finite generating set of a group is a big open problem. The conjecture is that they don't. –  Mark Sapir Jan 16 '11 at 22:10

Perhaps I can summarize my comments in this answer. The Ising model on Cayley graphs is similar but more difficult than percolation models (site, bond and others). The "canonical" reference for percolation on transitive graphs (including the Cayley graphs) is Benjamini-Schramm (see references here). There are deep connections between geometric properties of Cayley graphs (say, the growth, amenability, etc.) and properties of percolation (say, the number of different phases, critical exponents, etc). Similar connections exist for Ising model although that model is more difficult and much less explored. See references in Spakulova's thesis "Percolation and Ising Model on Tree-Like Graphs" and "The Ising model and percolation on trees and tree-like graphs" by Russell Lyons, Comm. Math. Phys. Volume 125, Number 2 (1989), 337-353.

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Hi Marcin,

recently writing a paper about phase transition on the Ising model with positive non-uniform magnetic field in infinite graphs I discovered that joining some results in the literature we can relate amenability and Phase transition in the Ising model with positive magnetic field:

Theorem 1: If $G$ is a non-amenable infinite connected graph then there is a ferromagnetic Ising model on $G$, with constant positive magnetic field having phase transition.

There is a partial converse of this result for quasi-transitive amenable graphs. A infinite graph $G=(V,E)$ is quasi-transitive if there exist a finite number of vertices $x_1,\ldots,x_k$ such that for any $x\in V$, there is an automorphism of $G$ taking $x$ to some $x_i$.

Theorem 2: If $G$ is a amenable quasi-transitive infinite connected graph then all ferromagnetic Ising model on $G$, with constant positive magnetic field has no phase transition.

Theorem 2 is interesting converse because of the quasi-transitive hypothesis can not be removed since Bausev shown that we have phase transition in ferromagnetic Ising model with magnetic field being constant at all sites of the lattice $\mathbb{Z}^2 \times \mathbb{Z}_+$ .

Definitions.
An Ising model on a graph $G=(V,E)$ is defined as follows:

Let $\mathcal{L}$ be the set of finite parts of $V$ and suppose that $\Lambda_n\in\mathcal{L}$ is such that $\cup_{n\in\mathbb{N}}\ \Lambda_n=G$. The Hamiltonian of the Ising model in $\Lambda_n$ with a boundary condition $\omega\in \{-1,1\}^{V}$ is given by $$H_{\Lambda_n}(\sigma|\omega)= -\sum_{\substack{i,j\in E:\\ i,j\in\Lambda_n}} J\ \sigma_i\sigma_j -\sum_{i\in \Lambda_n}h_i \ \sigma_i - \sum_{\substack{i,j\in E:\\ i\in\Lambda_n,j\in\Lambda_n^c}} J\ \sigma_i\omega_j,$$ where $\sigma=(\sigma_i)_{i\in V}\in\{-1,1\}^{V}$, $J\in\mathbb{R}$ (the model is called ferromagnetic of $J>0$) and $h_i\in\mathbb{R}$ is said the magnetic field. Finally we say that this Ising model has phase transition if the closed convex hull of the set $$\left\{w-\lim_{\Lambda_n\uparrow G}\ \mu_{\Lambda_n}^{\beta,\omega}:\omega\in\{-1,1\}^{V} \right\}$$ is singleton for all $\beta>0$. The measures $\mu_{\Lambda_n}^{\beta,\omega}$ are defined by $$\mu_{\Lambda_n}^{\beta,\omega}(\sigma)= \left\{ \begin{array}{rl} \frac{\exp(-\beta H_{\Lambda_n}^{\omega}(\sigma))}{Z_{\Lambda_n}^{\omega}},&\text{if}\ \sigma_i=\omega_i\ \forall i\in\Lambda_n^c;\\ 0,& \text{otherwise}, \end{array} \right.$$

One nice reference about the results I stated above is Jonasson, J. and Steif, J. E.: Amenability and Phase Transition in the Ising Model. J. Theor. Probab. 12, 549-559 (1999).

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This has been studied. See

@article {MR1390236, AUTHOR = {Regge, Tullio and Zecchina, Riccardo}, TITLE = {Exact solution of the {I}sing model on group lattices of genus {$g>1$}}, JOURNAL = {J. Math. Phys.}, FJOURNAL = {Journal of Mathematical Physics}, VOLUME = {37}, YEAR = {1996}, NUMBER = {6}, PAGES = {2796--2814}, ISSN = {0022-2488}, CODEN = {JMAPAQ}, MRCLASS = {82B20 (82B23)}, MRNUMBER = {1390236 (97f:82014)}, MRREVIEWER = {Gunter Sch{\"u}tz}, DOI = {10.1063/1.531690}, URL = {http://dx.doi.org/10.1063/1.531690}, }

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I can also add the PhD thesis of my former student Iva Spaculova. She studied the Ising model on groups with infinitely many ends, say, free groups (and arbitrary generating sets). It is more complicated (but somewhat similar) to percolation. See Iva's paper Špakulová, Iva, Critical percolation of virtually free groups and other tree-like graphs. Ann. Probab. 37 (2009), no. 6, 2262–2296. –  Mark Sapir Jan 16 '11 at 22:07
@Mark: thanks for the reference. I would have actually thought that the physicists would have looked at free groups quite early on ("Bethe Lattice") –  Igor Rivin Jan 17 '11 at 0:01
Physicists considered Ising model and percolation on trees (that is called the Bethe lattice). Cayley graphs of free groups with respect to arbitrary generating sets are much more complicated. –  Mark Sapir Jan 17 '11 at 3:36