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A renormalization group compares models at different observation lengths. A Wilsonian coarse-grained statistical Hamiltonian sums over short-distance fluctuations; a renormalization-group transformation rescales coordinates and fields for comparison. Kenneth Geddes Wilson's 1982 lecture treats fluctuations across successive length scales to explain critical phenomena.^[ref]

The experimental motivation is Andrews's carbon dioxide experiment. Dried carbon dioxide occupied a capillary tube above mercury; a screw compressed the transmitting water. A second air-filled tube gauged pressure. Flowing water controlled temperature, and a cathetometer read column positions for volume. The liquid meniscus vanished near the liquid-gas critical point. Residual air and imperfect air-gauge corrections limited accuracy.^[ref] These observations motivate critical-point theory without determining a correlation-length critical exponent. The renormalization group explains how critical measurements can become insensitive to many molecular details.

In 1972, Kenneth Geddes Wilson and Michael Ellis Fisher used an epsilon expansion about $d=4$ to compute nonclassical critical exponents. Their paper distinguishes a Gaussian fixed point from an interacting renormalization-group fixed point, tunes the thermal parameter, and extracts powers from a linearized recursion. Its Ising-like result is $\gamma =1+ϵ/6+O({ϵ}^2)$ for $d=4-ϵ$.^[ref] The worked model below isolates this mechanism with chosen coefficients; it is not a precision calculation for the three-dimensional Ising model.