  Polymer gels such as jellies and soft contact lenses are soft solids
consisting of three-dimensional polymer networks swollen with a large amount of
solvent. For approximately 80 years, the swelling of polymer gels has been
described using the Flory--Huggins mean-field theory. However, this theory is
problematic when applied to polymer gels with large solvent contents owing to
the significant fluctuations in polymer concentration. In this study, we
experimentally demonstrate the superiority of the semidilute scaling law over
the mean-field theory for predicting the swelling of polymer gels. Using the
semidilute scaling law, we experimentally determine the universal critical
exponent $\nu$ of the self-avoiding walk via swelling experiments on polymer
gels. The experimentally obtained value $\nu\simeq 0.589$ is consistent with
the previously reported value of $\nu\simeq 0.588$, which was obtained by
precise numerical calculations. Furthermore, we theoretically derive and
experimentally demonstrate a scaling law that governs the equilibrium
concentrations. This scaling law contradicts the predictions made by de Gennes'
$c^{*}$ theorem. A major deficiency of the $c^*$ theorem is that the network
elasticity, which depends on the as-prepared state, is neglected. These
findings reveal that the semidilute scaling law is a fundamental principle for
accurately predicting and controlling the equilibrium swelling of polymer gels.
