Let $\beta>0$. The Curie-Weiss Model of ferromagnetism is the probability distribution defined as follows. For $n \in \mathbb{N}$, define random variables $S_{1}, \ldots, S_{n}$ with values in $\{\pm 1\}$ such that the probabilities are given by

$$\mathbb{P}\left(S_{1}=s_{1}, \ldots, S_{n}=s_{n}\right)=\frac{1}{Z_{n, \beta}} \exp \left(\frac{\beta}{2 n} \sum_{i=1}^{n} \sum_{j=1}^{n} s_{i} s_{j}\right)$$

where $Z_{n, \beta}$ is the normalisation constant

$$Z_{n, \beta}=\sum_{s_{1} \in\{\pm 1\}} \cdots \sum_{s_{n} \in\{\pm 1\}} \exp \left(\frac{\beta}{2 n} \sum_{i=1}^{n} \sum_{j=1}^{n} s_{i} s_{j}\right)$$

(a) Show that $\mathbb{E}\left(S_{i}\right)=0$ for any $i$.

(b) Show that $\mathbb{P}\left(S_{2}=+1 \mid S_{1}=+1\right) \geqslant \mathbb{P}\left(S_{2}=+1\right)$. [You may use $\mathbb{E}\left(S_{i} S_{j}\right) \geqslant 0$ for all $i, j$ without proof. ]

(c) Let $M=\frac{1}{n} \sum_{i=1}^{n} S_{i}$. Show that $M$ takes values in $E_{n}=\left\{-1+\frac{2 k}{n}: k=0, \ldots, n\right\}$, and that for each $m \in E_{n}$ the number of possible values of $\left(S_{1}, \ldots, S_{n}\right)$ such that $M=m$ is

$$\frac{n !}{\left(\frac{1+m}{2} n\right) !\left(\frac{1-m}{2} n\right) !}$$

Find $\mathbb{P}(M=m)$ for any $m \in E_{n}$.