We consider in detail the critical behavior of a d-dimensional system that is finite in one of its dimensions. Such a system can exhibit four different types of critical behavior: d-dimensional mean field, d-dimensional critical, (d-1)-dimensional mean field, (d-1)-dimensional critical. By matching its behavior in the various regions we obtain equalities connecting the critical indices in d and d-1 dimensions. Assuming the usual two-dimensional critical indices of the Ising model, these relations lead to the following indices for the three-dimensional Ising model: ν=2/3, α=0, β=3/8, and γ=5/4.
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Imry et al. (1973) studied this question.
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