Why do we study the Ising model on $\mathbb{Z}^d$ for $d > 3$? I'm a beginner in statistical physics and I'm reading some stuff about the Ising model. So this might be a silly question. My question is: why we study the Ising model for high dimension cases, despite that our physical world has only dimension $2$ or $3$?
 A: Let us start with a quote from a paper by Michael Fisher and David Gaunt in 1964 (Phys. Rev. 133, A224), at a time when it was still necessary to justify such studies:

To elucidate the general problem of dependence on dimensionality and coordination number, it seemed worthwhile to investigate the Ising model and self-avoiding walks for lattices of dimensionality higher than three. [...] Of course the behavior of physical systems in four or more space-like dimensions is not directly relevant to comparison with experiment! We can hope, however, to gain theoretical insight into the general mechanism and nature of phase transitions.

As they say, it turns out that the spatial dimensionality (and more generally the connectivity properties of the underlying graph) plays a major role in the behavior of macroscopic systems. This is certainly the case at a critical point, where the critical exponents are well-known to depend generally on the spatial dimension, but can also be seen away from the critical point. As one example of the latter, consider the asymptotic behavior of the energy-energy correlations above the critical temperature:
$$
\langle \epsilon_0\epsilon_{n\vec e_1} \rangle_\beta - \langle \epsilon_0\rangle_\beta \langle \epsilon_{n\vec e_1} \rangle_\beta \sim
\begin{cases}
n^{-2} e^{-2n/\xi}    &   (d=2)\\
n^{-2}(\log n)^{-2} e^{-2n/\xi}   &   (d=3)\\
n^{-(d-1)} e^{-2n/\xi}    &   (d\geq 4)
\end{cases}
$$
where, for any $k\in\mathbb{Z}^d$, $\epsilon_k = \sigma_k\sigma_{k+\vec{e}_1}$ and $\xi$ denotes the correlation length. As can be seen, the corrections to the exponential decay exhibit an interesting dependence on the dimension, which it is very natural for a physicist to try to understand.
