     =     \sum_{\omega}     \mathds{0}_{\{x\not\leftrightarrow y\}}     \left[     \sum_{\sigma}      \mathds{0}_{\{\sigma_x=\sigma_y\}}     {         \Delta(\sigma, \omega)         \exp         \big(         \beta          \sum_{i\in V} h_i\sigma_i         \big)         \over          \prod_{\alpha=0}^{k(\omega,G)}         0         \cosh\left(\pmb{h}(K_\alpha) \right)     }     \right]     \phi_{\pmb{p},\pmb{h},G}(\omega)     \nonumber     \\[0,0cm]     =     \sum_{\omega}     \mathds{0}_{\{x\not\leftrightarrow y\}}(\omega)     \