= Solution
The <renormalization group> organizes how a statistical system changes when observed at progressively larger length scales. Start with its <partition function> and a microscopic cutoff $\Lambda$. Split the <Fourier transform> field into modes below $\Lambda/b$ and modes in the shell $\Lambda/b<|q|<\Lambda$. Integrate the shell modes, rescale coordinates $x'=x/b$ to restore the cutoff, and rescale the field to keep a chosen gradient normalization. This is a <renormalization-group transformation>. It preserves the <partition function> when the generated interactions and additive <free energy> terms are retained; truncating the resulting operator expansion is an approximation.
A <renormalization-group fixed point> has an invariant dimensionless interaction structure. The continuous flow of a nearby scaling field $v_i$ takes the linearized form
$$
\frac{dv_i}{d\ell}=y_iv_i+O(v^2),\qquad \ell=\log b.
$$
A positive $y_i$ defines a <relevant operator>, a negative $y_i$ an <irrelevant operator>, and a zero $y_i$ a <marginal operator> requiring nonlinear analysis. The continuous-flow eigenvalues $y_i$ should not be confused with the discrete multipliers $b^{y_i}$. To reach a critical point, tune the relevant thermal variable and set the relevant symmetry-breaking source to zero. The resulting flow stays on the <critical surface>.
Microscopically different systems whose remaining couplings approach the same fixed point share a <universality class>. Irrelevant perturbations lose their influence on long-distance exponents. Relevant directions explain why varying the temperature or field moves the system away from criticality. The thermal scaling field obeys $t'=b^{y_t}t$, while the rescaled <correlation length> is $\xi'=\xi/b$. Taking $b=|t|^{-1/y_t}$ therefore gives $\nu=1/y_t$.
If the field has scaling dimension $x_\phi=(d-2+\eta)/2$, the source term $\int h\phi$ has eigenvalue $y_h=d-x_\phi=(d+2-\eta)/2$. The singular <free-energy density> then obeys
$$
f_s(t,h)=b^{-d}f_s(b^{y_t}t,b^{y_h}h),\qquad
2-\alpha=d/y_t,\qquad\Delta=y_h/y_t,
$$
when the <hyperscaling relation> is valid. This derives the <scaling hypothesis for critical phenomena> and the exponent relations of Question 2 from the fixed-point picture.
At the <Gaussian fixed point>, keeping $\int(\nabla\phi)^2$ invariant gives $x_\phi=(d-2)/2$. The quadratic coupling has $y_r=2$ and the quartic coupling has $y_u=4-d$. Thus quartic interactions are relevant below four dimensions, marginal at four, and irrelevant above four. Below four, in dimensions where an ordinary continuous short-range scalar transition exists, an interacting <Wilson-Fisher fixed point> governs that transition. Above four, the Gaussian fixed point gives mean-field powers, but the quartic term remains necessary to stabilize the ordered state: it is a <dangerously irrelevant coupling>, explaining the failure of naive <hyperscaling>. \b[The renormalization group connects scale invariance, universality, relevant tuning parameters and fluctuation corrections in one framework.]
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