Finite-dimensional persistence of the large-d obstruction

Determine whether the mechanism that obstructs smooth continuation of the perturbative renormalization-group fixed point from \(\varepsilon=0\) to the physical value \(\varepsilon=2\) in the large-d limit persists at finite spatial dimension.

Background

The paper’s evidence for an obstruction to continuing the perturbatively accessible real, stable RG fixed point is obtained at leading order in the $1/d$ expansion. The authors emphasize that the finite-dimensional case, especially the physically relevant case d=3d=3, is not currently resolved because only the first two orders of its ε\varepsilon-expansion are known.

A finite-dimensional test would require substantially higher-order calculations in the full stochastic Navier–Stokes theory. Establishing persistence of the mechanism would determine whether the large-d scenario is relevant to physical turbulence rather than being a peculiarity of the large-d limit.

References

This conclusion is obtained in the limit $d\to\infty$, and whether the same mechanism persists at finite $d$ remains an open question.

— Evidence for an Obstruction to Perturbative Renormalization Group in Stochastic Turbulence  (2610.01507 - Adzhemyan et al., 1 Oct 2026) in Discussion section

Constructing such an expansion without relying on the $\varepsilon$ parameter would, however, require a reformulation of the theory, which remains an open problem.

— Evidence for an Obstruction to Perturbative Renormalization Group in Stochastic Turbulence  (2610.01507 - Adzhemyan et al., 1 Oct 2026) in Discussion section