I Understanding the Derivation of the Ginzburg Criterion for the Ising Model

thatboi
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For the Ising Model, the Ginzburg Criterion is, for ##m_{0}## the order parameter and ##\delta m## the fluctuations: $$\langle\delta m\left(x\right)\delta m\left(x^{\prime}\right)\rangle << m_{0}^{2}$$. I want to understand how to derive the left hand side of the inequality from ##\langle M^{2} \rangle - \langle M \rangle ^{2}## where ##M = m_{0} + \delta_{m}##. Just from plugging in, I'm not sure how most of the terms cancel out, or what the fate of a term like ##\langle \delta m\rangle \langle \delta m \rangle## is.
 
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Ok, I think the best way of understanding that ##\langle \delta m \rangle = 0## is by calculating the probability associated with ##\delta m## i.e. calculating its Boltzmann weight and noticing that it is a Gaussian random variable with 0 mean. The other more physical way of arguing it is that we know ##m_{0}## is the value of the order parameter that minimizes the Helmholtz Free Energy so we expect ##\langle M \rangle = m_{0}##.
 
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