  We study model spaces, in the sense of Hairer, for stochastic partial
differential equations involving the fractional Laplacian. We prove that the
fractional Laplacian is a singular kernel suitable to apply the theory of
regularity structures. Our main contribution is to study the dependence of the
model space for a regularity structure on the three-parameter problem involving
the spatial dimension, the polynomial order of the nonlinearity, and the
exponent of the fractional Laplacian. The goal is to investigate the growth of
the model space under parameter variation. In particular, we prove several
results in the approaching subcriticality limit leading to universal growth
exponents of the regularity structure. A key role is played by the viewpoint
that model spaces can be identified with families of rooted trees. Our proofs
are based upon a geometrical construction similar to Newton polygons for
classical Taylor series and various combinatorial arguments. We also present
several explicit examples listing all elements with negative homogeneity by
implementing a new symbolic software package to work with regularity
structures. We use this package to illustrate our analytical results and to
obtain new conjectures regarding coarse-grained network measures for model
spaces.
