# Rationalising the 1/B scaling

From [Firth (1993)](https://www.jstor.org/stable/2336755#metadata_info_tab_contents) we know that the biased estimator for one of the Ising coupling asymptotically goes like

$$J_{ij}=\bar{J}_{ij}+\frac{\tilde{B}_{ij}}{B}$$

where $\bar{J}_{ij}$ is the unbiased (true) value. So the average will be

$$\mathbb{E}[J]=\frac{2}{N(N-1)}\sum_{i<j}J_{ij}=\bar{J}+\tilde{B}/B$$

while the variance is

$$\rm{Var}[J]=\mathbb{E}[J^2]-\mathbb{E}[J]^2$$

Every $J^2_{ij}$ is

$$J_{ij}^2=\bar{J}_{ij}^2+2\bar{J}_{ij}\frac{\tilde{B}_{ij}}{B}+\left(\frac{\tilde{B}_{ij}}{B}\right)^2$$

so its leading order is in $1/B$. Hence, the leading order for the variance comes from the cross product term, so that

$$\rm{Var}[J]=C+\frac{D}{B}+\dots$$

The standard deviation is then

$$\sigma = \sqrt{\rm{Va}r[J]}$$

and the "temperature" is

$$T=\dfrac{1}{\sigma}=\dfrac{1}{\sqrt{C}} -\dfrac{D}{2 C^{3/2} B}+\dots= \bar{T}-\dfrac{D \bar{T}^3}{2B}$$