A coupling direction is <relevant coupling>[relevant], <irrelevant coupling>[irrelevant], or <marginal coupling>[marginal] according as its renormalization-group eigenvalue is positive, negative, or zero. The <critical surface> is the stable manifold of a critical fixed point: initial couplings on it flow toward that fixed point, while relevant perturbations take the theory away from criticality. Microscopic models with different irrelevant couplings lose those distinctions under coarse-graining and approach the same fixed point, which explains their shared long-distance behavior and <universality class>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 303 / 2 / a / iii / Solution

{wiki=Renormalization_group#Critical_surface}

The critical surface is the stable manifold of a critical <renormalization-group fixed point>. Tuning every relevant direction places a theory on this surface.

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 Incoming links: Critical surface