Near the ordinary critical point, $K\sim at+\gamma k^2$. Rescaling $k=t^{1/2}q$ gives the <Gaussian fluctuation correction near a critical point>
$$
\int\frac{d^dk}{(t+k^2)^2}\asymp t^{d/2-2}.
$$
Thus $\alpha_{\mathrm{fl}}=2-d/2$. It becomes as important as the mean-field value $\alpha=0$ at $d=4$, so the ordinary <upper critical dimension> is $d_c=4$.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 303 / 1 / b / iii / Solution

For a quadratic kernel $K(t,k)\sim t+k^2$, the singular heat-capacity correction is proportional to
$$
\int\frac{d^dk}{(t+k^2)^2}\asymp t^{d/2-2}.
$$


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