For which values of $N$ is known the Lieb-Simon Inequality for $Z_N$ Models ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T16:47:14Z http://mathoverflow.net/feeds/question/27549 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/27549/for-which-values-of-n-is-known-the-lieb-simon-inequality-for-z-n-models For which values of $N$ is known the Lieb-Simon Inequality for $Z_N$ Models ? Leandro 2010-06-09T04:24:25Z 2010-11-24T12:22:13Z <p><strong> Background:</strong></p> <p>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 <code>$\{\exp(2\pi i j/N);1\leq j\leq N\}$</code> with uniform a priori distribution. </p> <p>Consider the formal Hamiltonian given on the lattice $\mathbb Z^d$ by $$H_{ \Lambda }({\sigma}) = -\sum_{\langle x,y\rangle} J_{xy} \vec\sigma_{x} \vec\sigma_{y}<br>$$ 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<br> $\Lambda\subset \mathbb Z^d$ 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 $$\langle \vec\sigma_x \vec\sigma_y \rangle_{\Lambda}= Z_{\Lambda}^{-1}\int\vec\sigma_x \vec\sigma_y\exp\Big( \beta \sum_{\langle x,y\rangle \in \Lambda }J_{xy} \vec\sigma_x \vec\sigma_y\Big) d\sigma$$ <strong> Question:</strong></p> <p>For which values of $N$ is known that the Lieb-Simon Inequality is true or false ?</p> <p><strong> Lieb-Simon Inequality</strong> $$\langle \vec\sigma_x \vec\sigma_y \rangle_{\Lambda} \leq \sum_{b \in \partial B} \bigl&lt;\vec\sigma_x \vec\sigma_b \bigr>_B \bigl&lt;\vec\sigma_b \vec\sigma_y \bigr>_\Lambda,$$ where $B\subset\Lambda\subset\mathbb Z^d$ are finite, $x,y\in\Lambda$, <code>$\partial B=\{z\in B; d(z,B^c)=1\}$</code> and $\partial B$ separates $x$ and $y$ ($i.e.$ any path from $x$ to $y$ must intercept $\partial B$).</p> http://mathoverflow.net/questions/27549/for-which-values-of-n-is-known-the-lieb-simon-inequality-for-z-n-models/27561#27561 Answer by Yvan Velenik for For which values of $N$ is known the Lieb-Simon Inequality for $Z_N$ Models ? Yvan Velenik 2010-06-09T08:21:48Z 2010-06-09T08:21:48Z <p>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.</p>