= Solution
A <phase diagram> locates stable <pure thermodynamic phases> and their <phase coexistence> boundaries in a space of control parameters. For a concrete three-dimensional example, take the temperature, crystal-field parameter, and magnetic field of the <Blume–Capel model>. Its spins take values $0,\pm1$; changing the cost of the zero-spin state can turn a continuous ordering transition into a discontinuous one. A local scalar description near the resulting <tricritical point> is the <Landau-Ginzburg theory>
$$
\mathcal F[M]=\int d^Dx\left[\frac K2|\nabla M|^2+V(M)\right],\qquad
V(M)=\frac r2M^2+\frac u4M^4+\frac v6M^6-hM,\quad K,v>0.
$$
The three local control coordinates are $(r,u,h)$, with $r$ varying linearly with the thermal scaling field. For $u>0$, $h=0$, the transition at $r=0$ is continuous. For $u<0$, zero-field coexistence is at $r=3u^2/(16v)$, and the <magnetization> jumps between zero and $\pm\sqrt{-3u/(4v)}$. The two loci meet at $r=u=h=0$, the <tricritical point>.
The full three-dimensional <phase diagram> has an ordered coexistence sheet at $h=0$, where the two opposite magnetizations exchange stability when $h$ changes sign. On its three-phase line it meets two nonzero-field <tricritical wings>. These wings are <first-order phase transition> surfaces between minima of the same sign but different magnitude, terminating at ordinary <thermodynamic critical points>. Their <tricritical wing critical edges> are found by $V'=V''=V'''=0$:
$$
M_c^2=-\frac{3u}{10v},\qquad r_c=\frac{9u^2}{20v},\qquad h_c=\frac{6u^2M_c}{25v}\quad(u<0).
$$
Indeed $V'''=6uM+20vM^3$ fixes $M_c^2$, and substituting into $V''=0$ and $V'=0$ gives $r_c,h_c$; $V''''=-12u>0$ ensures a stable critical quartic minimum. The two signs give the two wings. All transition surfaces and critical edges end together at the <tricritical point>. The following plot uses $v=1$; its wing surfaces come from the exact <tricritical wing coexistence factorization>, rather than schematic unconstrained surfaces.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/iii/paper-50-landau.png]
{title=Tricritical coexistence surfaces, the zero-field phase diagram, and ordinary, first-order and tricritical Landau potentials}
The unheaded requests for <critical exponents> follow from the uniform equation of state
$$
h=rM+uM^3+vM^5.
$$
For an ordinary <thermodynamic critical point>, $u>0$ is fixed and $r=At$, $A>0$. At $h=0$, $t<0$, the small ordered minimum obeys $M^2\sim-r/u$, so the <order-parameter critical exponent> is $\beta=1/2$. Above the transition $\chi=(\partial M/\partial h)_{h=0}=1/r$, giving the <magnetic-susceptibility critical exponent> $\gamma=1$. At $r=0$, $h\sim uM^3$, so the critical-isotherm exponent is $\delta=3$. At the tuned <tricritical point>, $u=0$, instead $M^4=-r/v$ below the transition and $h=vM^5$ on its critical isotherm, while the disordered susceptibility remains $1/r$. Therefore
$$
\boxed{(\beta,\gamma,\delta)_{\rm ordinary}=(1/2,1,3),\qquad
(\beta,\gamma,\delta)_{\rm tricritical}=(1/4,1,5).}
$$
Below the transition the inverse susceptibility is $r+3uM^2+5vM^4$, giving $2|r|$ in the ordinary limit and $4|r|$ in the tricritical limit. Thus both sides give the same susceptibility power, with different amplitudes.
To derive the general <Ginzburg criterion>, tune away all lower even nonlinearities and let the leading stabilizer be $v_nM^{2n}/(2n)$, $n\geq2$. The equation of state gives $M_0^2\asymp|r|^{1/(n-1)}$, and its curvature at the ordered saddle is $2(n-1)|r|$, so $\xi\asymp|r|^{-1/2}$. The fluctuations of the <order parameter> averaged over a <correlation volume> are estimated by its long-wavelength quadratic covariance:
$$
\langle(\delta M)^2\rangle_\xi
\asymp k_BT\int_{|q|\lesssim\xi^{-1}}\frac{d^Dq}{(2\pi)^D}\frac1{Kq^2+c|r|}
\asymp\xi^{2-D}.
$$
The integral has a fixed dimensionless upper limit after $q=p/\xi$; this is a coarse-volume fluctuation estimate, not the ultraviolet-dominated point variance. Consequently the <multicritical Ginzburg ratio> is
$$
\frac{\langle(\delta M)^2\rangle_\xi}{M_0^2}
\asymp |r|^{(D-2)/2-1/(n-1)}.
$$
The <Landau approximation> is asymptotically controlled by this test only when the ratio tends to zero. This gives the <upper critical dimension>
$$
\boxed{D_c=\frac{2n}{n-1};\qquad D_c=4\ (n=2),\quad D_c=3\ (n=3).}
$$
The same threshold follows by giving $M$ its Gaussian <engineering dimension> $(D-2)/2$: the coupling to $M^{2n}$ has scaling exponent $D-n(D-2)=2n-(n-1)D$, which becomes relevant below $D_c$.
For $D<D_c$, the fluctuation ratio diverges and the saddle calculation cannot predict the generic interacting <critical exponents>. At $D=D_c$ the test is marginal, not a proof of changed power exponents: the ratio has zero thermal power and marginal interactions can produce logarithmic corrections. Thus the PDF's inclusion of equality in a blanket claim about wrong power exponents needs this qualification. The <Landau-Ginzburg theory> as a fluctuating field theory remains useful; what loses validity is its fluctuation-free <Landau approximation>.
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