## 4 Analysis of DoF and User Scaling
### Interference Decaying Rate  
Next, we analyze the average interference decaying rate defined as the average decaying rate of the total amount of received interference and/or generating interference with respect to the number of MSs, \(N\). This is meaningful since the desired user scaling law is closely related to the interference decaying rate with increasing \(N\) for given SNR.  
Let \(\mathcal{I}_{\text{min},M}^{(d)}\) denote the maximum value (i.e., the \(M\)th smallest value) among the downlink interference levels that \(M\) selected downlink MSs compute, which is given by  
\[\mathcal{I}_{\text{min},M}^{(d)}=\max_{\pi_{m}\in\mathcal{S}^{(d)}}L_{\pi_{m}}, \tag{24}\]  
where \(L_{\pi_{m}}\) represents the downlink scheduling metric of selected MS \(\pi_{m}\) and \(\mathcal{S}^{(d)}\) is the set of selected downlink MSs. In addition, let \(\mathcal{I}_{\text{min},M}^{(u)}\) denote the maximum value among the MS-to-MS interference levels that \(M\) selected uplink MSs compute, which is given by  
\[\mathcal{I}_{\text{min},M}^{(u)}=\max_{\phi_{j}\in\mathcal{S}^{(u)}}\gamma_{ \phi_{j}}, \tag{25}\]  
where \(\gamma_{\phi_{j}}\) is the uplink scheduling metric of selected MS \(\phi_{j}\) as shown in Equation (6) and \(\mathcal{S}^{(u)}\) is the set of selected uplink MSs. Since the performance of our hybrid opportunistic scheduling method is limited mainly by (1) such a selected downlink MS that receives the maximum amount of interference from other beams generated by the BS or (2) such a selected uplink MS that generates the maximum amount of interference to selected downlink MSs, it is certainly worth analyzing an asymptotic behavior of \(\mathcal{I}_{\text{min},2M}\triangleq\max\{\mathcal{I}_{\text{min},M}^{(d)}, \mathcal{I}_{\text{min},M}^{(u)}\}\) with respect to \(N\).  
Now, we are ready to establish our second main result, which shows a lower bound on the average interference decaying rate \(\mathbb{E}\left[\frac{1}{\mathcal{I}_{\text{min},2M}}\right]\) with respect to \(N\).  
Theorem 4.2: _For the multi-antenna full-duplex system in Section 2, the average interference decaying rate is lower-bounded by_  
\[\mathbb{E}\left[\frac{1}{\mathcal{I}_{\text{min},2M}}\right]\geq\Theta\left( N^{1/M}\right). \tag{26}\]  
Proof: The proof essentially follows the same steps as those in (Section III-B in [32]) and (Remark 1 in [26]), and thus a brief sketch of the proof is provided here. From the proof of Theorem 4.1 and the [PERSON]'s inequality, it follows that  
\[\begin{split} 1-\Pr\left\{\mathcal{I}_{\text{min},2M}\leq\frac{ \epsilon}{\text{SNR}}\right\}&\leq\frac{M\cdot\text{SNR}}{ \epsilon}\mathbb{E}\left[\max\{\mathcal{I}_{\text{min},M}^{(d)},\mathcal{I}_{ \text{min},M}^{(u)}\}\right]\\ &=\frac{M\cdot\text{SNR}}{\epsilon}\mathbb{E}\left[\max\left\{ \max_{\pi_{m}\in\mathcal{S}^{(d)}}L_{\pi_{m}},\max_{\phi_{j}\in\mathcal{S}^{(u )}}\gamma_{\pi_{j}}\right\}\right]\\ &=\Theta\left(\frac{\text{SNR}}{N^{1/M}}\right)\end{split} \tag{27}\]  
for small \(\epsilon>0\), which tends to zero if \(N=\omega\left(\text{SNR}^{M}\right)\). Here, the first equality holds due to Equations (24) and (25). This completes the proof of Theorem 4.  
From the above theorem, we obtain the same scaling law as in Theorem 1. This implies that the faster interference decaying rate with respect to \(N\), the smaller SNR exponent in the user scaling law.