Paper 2, Section II, F

Probability | Part IA, 2017

Let β>0\beta>0. The Curie-Weiss Model of ferromagnetism is the probability distribution defined as follows. For nNn \in \mathbb{N}, define random variables S1,,SnS_{1}, \ldots, S_{n} with values in {±1}\{\pm 1\} such that the probabilities are given by

P(S1=s1,,Sn=sn)=1Zn,βexp(β2ni=1nj=1nsisj)\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 Zn,βZ_{n, \beta} is the normalisation constant

Zn,β=s1{±1}sn{±1}exp(β2ni=1nj=1nsisj)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 E(Si)=0\mathbb{E}\left(S_{i}\right)=0 for any ii.

(b) Show that P(S2=+1S1=+1)P(S2=+1)\mathbb{P}\left(S_{2}=+1 \mid S_{1}=+1\right) \geqslant \mathbb{P}\left(S_{2}=+1\right). [You may use E(SiSj)0\mathbb{E}\left(S_{i} S_{j}\right) \geqslant 0 for all i,ji, j without proof. ]

(c) Let M=1ni=1nSiM=\frac{1}{n} \sum_{i=1}^{n} S_{i}. Show that MM takes values in En={1+2kn:k=0,,n}E_{n}=\left\{-1+\frac{2 k}{n}: k=0, \ldots, n\right\}, and that for each mEnm \in E_{n} the number of possible values of (S1,,Sn)\left(S_{1}, \ldots, S_{n}\right) such that M=mM=m is

n!(1+m2n)!(1m2n)!\frac{n !}{\left(\frac{1+m}{2} n\right) !\left(\frac{1-m}{2} n\right) !}

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

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