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I am studying a $d$-state Potts model. A configuration $\sigma$, which assigns for each $x\in \mathbb{Z}^2$ a value $\sigma(x)\in [1,2,\ldots,d]$, with the probability on a finite lattice defined as $$ P_{\Lambda}(\sigma) = \frac{1}{\bf Z_{\Lambda}} e^{-H_{\Lambda}(\sigma)}, $$ where $H_{\Lambda}(\sigma)$ is a Hamiltonian and $\bf Z$ is a normalising constant. The form of the Hamiltonian is

$$ H_{\Lambda}(\sigma) = -\beta\sum_{\{d_{\infty}(x,y)=1\} \cap\{ y\in \Lambda\}\cap\{ x\in \Lambda\}} \delta\bigl(\sigma(x),\sigma(y)\bigr) + \sum_{x\in \Lambda} \sum_{i=1}^d h_{x,i} \delta\bigl(i,\sigma(x)\bigr), $$ where $x \in \Lambda$ where $\Lambda$ is a regular lattice in $ \mathbb{Z}^2,\beta>0$ and $h_{x,i}\geq h>0$. The function $\delta$ is the (Kronecker?) delta, that is $$ \delta(a,b) = \begin{cases} 1\, \mbox{ if } a=b, \\ 0\, \mbox{ otherwise}. \end{cases} $$

What I am wondering, is it possible to bound the absolute value of the covariance function (correlation function in physics?) $$\lvert\langle\delta(\sigma(x),i),\delta(\sigma(y),j)\rangle - \langle\delta(\sigma(x),i)\rangle \langle\delta(\sigma(y),j)\rangle\rvert,$$ by an exponential function if $d(x,y)$ is large? (depending only on $h,\beta$ and $d(x,y)$)

If not is the rate of decay known?

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    $\begingroup$ Sorry, but what are $ \delta $ and $ \sigma $ ? $\endgroup$ Oct 6, 2013 at 0:42
  • $\begingroup$ I echo what Emilio said, there is some notation that could use clarification, but after that this seems to be a good question. :-) $\endgroup$
    – David Z
    Oct 6, 2013 at 3:12
  • $\begingroup$ Sorry about the unclarity, I have expanded the description; is it clearer now? $\endgroup$ Oct 6, 2013 at 7:55

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