# Detection of Transmission State of Multiple Wireless Sources: A Statistical Mechanics Approach
## 4 Connection With Statistical Mechanics
This can be achieved directly by taking the limit \(\beta\rightarrow\infty\), i.e.,  
\[\widehat{\mathbf{s}}=\operatorname*{argmin}_{\mathbf{s}\in\{-1,+1\}^{N}} \mathcal{E}=-\lim_{\beta\rightarrow\infty}\frac{1}{N\beta}\ln(\mathcal{Z}) \tag{32}\]  
Similar arguments to the above can be also made if we generalize the domain of the state variables \(s_{i}\) from binary to continuous. The free energy is a random quantity depending on the realization of the network and the noise. However, it has been conjectured [50] (and proven in some cases) that it self-averages in the large system limit in the sense that  
\[\lim_{N\rightarrow\infty}\mathcal{F}_{N}=\lim_{N\rightarrow\infty}\mathbb{E }[\mathcal{F}_{N}] \tag{33}\]  
If \(\mathcal{E}\) depends on external \"quenched\" random variables, i.e., \(\mathcal{E}(s,\mathrm{G})\), then \(\mathcal{E}_{\min}\) and \(\mathcal{F}\) also depend on \(\mathrm{G}\). To evaluate the quantity \(\mathbb{E}[\ln(\mathcal{Z})]\), where the expectation is taken with respect to \(\mathrm{G},y,T,R\), we perform the replica trick, which can be expressed as  
\[\mathbb{E}\Big{[}\ln(\mathcal{Z})\Big{]}=\lim_{n\to 0^{+}}\frac{1}{n}\ln \Big{(}\mathbb{E}[\mathcal{Z}^{n}]\Big{)} \tag{34}\]  
where it is assumed that the resulting expression holds when analytically continuing \(n\to 0^{+}\). In summary, we can, therefore, make the following assumptions:  
**Assumption 1** (Replica Assumption). : \(\mathbb{E}[\mathcal{Z}^{n}]\) _evaluated for \(n\in\mathbb{N}^{+}\) can be analytically continued in the vicinity of \(n\to 0^{+}\)._  
_This assumption has seen widespread use in the field of statistical mechanics for alleviating the problem of dealing with averages of logarithms of random quantities since the logarithm is obtained after calculating \(\mathcal{Z}^{n}\), as demonstrated in Appendix B._  
Figure 2: The graphical model we used for establishing the network structure of an SU network. Shaded nodes distribute the messages of the respective source to all the neighbors connected with it. \(s_{\mu}\) (\(\widehat{h}_{\rightarrow\mu}\)) and \(y_{i}\) (\({h}_{\rightarrow i}\)) are the incoming messages to the primary source and secondary user, respectively. The network structure has been modeled as a Bethe lattice [49], where every secondary user/primary source is connected to an arbitrary number of primary sources/secondary users. We have assumed, without loss of generality, that the secondary user \(y_{i}\) is the origin of the lattice and all the other sources are arranged in shells around this root node. **Assumption 2** (Interchanging Limits). : _The limits \(N\to\infty\) and \(n\to 0^{+}\) in evaluating \(\mathcal{Z}^{n}\) can be interchanged._  
_The above assumptions enable us to obtain an expression for \(\mathbb{E}[\mathcal{Z}^{n}]\), which we need to evaluate using the saddle-point approximation. Here, we need to guess the right symmetries for the correct saddle-point solution. To do this, we invoke an additional assumption._  
**Assumption 3** (Replica Symmetry). : _The correct saddle-point solution is symmetric under the permutation of replica indices._  
_Although quite appealing, this assumption is often incorrect. In fact, in several instances, it has been shown that the saddle-point solutions break replica symmetry [40]. Nevertheless, given the difficulty of evaluating the replica-symmetry-broken solution, it is customary in similar applications to proceed with the replica-symmetric assumption, which has been shown to yield results very close to the correct ones._