## For which values of $N$ is known the Lieb-Simon Inequality for $Z_N$ Models ?

Background:

Let $\mathbb Z^d$ denote the $d$-dimensional integer lattice with norm $|x|=\sum_i|x_i|$. For each $x\in\mathbb Z^d$ we associate a spin variable, $\sigma_x$ taking values on the set $\{\exp(2\pi i j/N);1\leq j\leq N\}$ with uniform a priori distribution.

Consider the formal Hamiltonian given on the lattice $\mathbb Z^d$ by $$H_{ \Lambda }({\sigma}) = -\sum_{\langle x,y\rangle} J_{xy} \vec\sigma_{x} \vec\sigma_{y}$$ where $J_{xy}$ are nonnegative constants and $\vec\sigma_x \vec\sigma_y$ is inner product in $\mathbb R^2$. The sum is taken over all pair of first neighbors $\langle x,y\rangle$ means that $|x-y|=1$. The Partition function on a finite
$\Lambda\subset \mathbb Z^d$ is given by $$Z_\Lambda = \int\exp\Big( \beta \sum_{\langle x,y\rangle \in \Lambda }J_{xy} \vec\sigma_x \vec\sigma_y\Big)\;d\sigma$$ where the integral is taken over all sites of $\Lambda$. The two point correlations are given by $$\langle \vec\sigma_x \vec\sigma_y \rangle_{\Lambda}= Z_{\Lambda}^{-1}\int\vec\sigma_x \vec\sigma_y\exp\Big( \beta \sum_{\langle x,y\rangle \in \Lambda }J_{xy} \vec\sigma_x \vec\sigma_y\Big) d\sigma$$ Question:

For which values of $N$ is known that the Lieb-Simon Inequality is true or false ?

Lieb-Simon Inequality $$\langle \vec\sigma_x \vec\sigma_y \rangle_{\Lambda} \leq \sum_{b \in \partial B} \bigl<\vec\sigma_x \vec\sigma_b \bigr>_B \bigl<\vec\sigma_b \vec\sigma_y \bigr>_\Lambda,$$ where $B\subset\Lambda\subset\mathbb Z^d$ are finite, $x,y\in\Lambda$, $\partial B=\{z\in B; d(z,B^c)=1\}$ and $\partial B$ separates $x$ and $y$ ($i.e.$ any path from $x$ to $y$ must intercept $\partial B$).

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 Is it possible to add statistical Mechanics tag for this question ? – Leandro Jun 9 2010 at 4:35 statistical-physics is an existing tag, so I added that instead. – Qiaochu Yuan Jun 9 2010 at 4:41 Thank you Qiaochu ! – Leandro Jun 9 2010 at 4:42

As far as I know, it has only been proved for $N=1$ (Lieb) and $N=2$ (Lieb+Rivasseau) in the form you want. With an additional prefactor $\beta/N$ and with the infinite-volume measure in the RHS, it has been extended to $N=3$ and $N=4$ by Aizenman and Simon (see also Spohn and Zwerger). All these proofs rely on various correlation inequalities that, to my knowledge, have not been extended to general $O(N)$ models.

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Hi Velenik, perhaps you thought about $N$ as the dimension where spin variable lives and in this case I agree with you about the results. But here I am using $N$ to denote the number of states. I miss to add that $N=2$ here is the Ising model, where the inequality holds and I also Know that is valid for the XY model as you mention (Lieb+Rivasseua) that would be something like the limit when $N$ goes to infinity. Because this last observation I was think about that could be reasonable to prove it for large $N$ at least. I also heard one time that it could not be valid for $N=3$. – Leandro Jun 9 2010 at 14:08
Because in $N=3$ this model should be equivalent to 3 state Potts model. But I never locate any reference proving this statement. – Leandro Jun 9 2010 at 15:24
Right. I misread your question, sorry. I'll think about it (no ref come immediately to mind, though). Would you settle for the weak form (infinite volume measure and possibly some ($\beta$ and $N$ dependent) prefactor?). It is pretty clear how to derive such a result for $\beta < \beta_c$ in Potts model, but I have no idea how to treat general values of $\beta$ (the graphical representations available don't seem nice enough, but I should think more). On the other hand, I would simply look at the proofs of Lieb, Rivasseau, etc., to see whether they can be extended to that case... – Yvan Velenik Jun 9 2010 at 18:43
Yes, I am doing that. But in this case it leads more challenge comparison between the multigraphs and I am not able to do that. For this $Z_N$ model we have to take in account oriented multigraphs and also the Euler circuits that rises in the simon proof now are more general because it is allowed the degree to be zero modulus $N$. If you remember the reference, please let me know. I will think about to post the multigraph problem here also. Thanks – Leandro Jun 9 2010 at 19:46
Actually, the strong form cannot hold for general $q$-states Potts models. Indeed, it would imply (see the paper by Lieb) that the massgap is continuous at the phase transition, a fact that is known not to hold for large values of $q$. Of course, one might still believe that (i) the strong form holds for small values of $q$ (which should be $q\leq 4$ in dimension $2$, but only $q=2$, i.e. the Ising model, when $d\geq 3$), or (ii) the weak form holds more generally... – Yvan Velenik Jun 10 2010 at 6:57
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