First question: Wiki says Ising Model higher than four dimensions can be described by mean field theory. What is the reason for this? Does this mean there is no phase transition for higher dimensions than four?

Second question: The transverse Ising model $$H = -J\sum_{ij} \sigma^z_i \sigma^z_j-h \sum_i\sigma^x_i$$

is the quantum correspondence of Ising model. Does the statement above hold for this model?

First question: Wiki says Ising Model higher than four dimensions can be described by mean field theory. What is the reason for this? Does this mean there is no phase transition for higher dimensions than four? I can see fluctuations become less important in higher dimensions since there are many "neighbors" to stabilize the interactions but I'm looking for a more quantitative argument.

Second question: The transverse Ising model $$H = -J\sum_{ij} \sigma^z_i \sigma^z_j-h \sum_i\sigma^x_i$$

is the quantum correspondence of Ising model. Does the statement above hold for this model?