For couplings $g_i(\ell)$ under coarse-graining scale $b=e^\ell$, their <beta function (physics)> is $\beta_i(g)=dg_i/d\ell$. At a <renormalization-group fixed point> $g_*$ all beta functions vanish. Linearization gives
$$
\frac{d}{d\ell}\delta g_i=M_{ij}\delta g_j,
\qquad
M_{ij}=\left.\frac{\partial\beta_i}{\partial g_j}\right|_{g_*}.
$$
The eigenvalues $y_a$ of the <stability matrix of a renormalization-group fixed point> are the scaling dimensions of the associated coupling directions: $\delta g_a(\ell)=e^{y_a\ell}\delta g_a(0)$.

Solution

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

The stability matrix is the Jacobian of the beta functions at a <renormalization-group fixed point>. Its eigenvalues classify perturbations as relevant, irrelevant, or marginal.

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