Dimension where dimensional reduction fails in the random-field Ising model
In plain words
Adding a random magnetic field to the Ising model was once predicted to make it behave like the pure model in two fewer dimensions. That prediction is wrong in three dimensions, and it is unsettled at which dimension it starts to fail and why.
Precise statement
For the random-field Ising model with Gaussian random fields, Parisi-Sourlas supersymmetry predicts that critical exponents in dimension d equal those of the pure Ising model in $d - 2$. Determine the dimension $d_{\mathrm{DR}}$ below which this dimensional reduction fails and the mechanism, a cusp in the renormalized disorder cumulants (functional RG, d_DR approximately 5.1) or new relevant operators (other RG analyses, $d_{\mathrm{DR}}\ \text{between}\ 4\ \text{and}\ 5$).
What would settle it
Zero-temperature ground-state simulations in $d = 4 \text{ and } 5$ testing supersymmetry relations between exponents, matched by an RG computation of $d_{\mathrm{DR}}$ with controlled error.
Status in the literature
Unverified note
Simulations find supersymmetry satisfied in $d = 5$ and violated in $d = 4$ (arXiv:1901.08473, 2019); a 2024-2025 functional RG study estimates $d_{\mathrm{DR}} = 5.11(9)$ (arXiv:2411.11147), in tension with other RG analyses that place $d_{\mathrm{DR}}\ \text{between}\ 4\ \text{and}\ 5$.