This thesis investigates the vector-valued Gaussian free field (GFF) and its applications to statistical mechanics, focusing on the analytically challenging spin O(N) model and the more tractable spherical model. Both are fundamental in mathematical physics and can be related to the GFF via local and global conditioning on its norm, respectively.

We begin by analyzing the discrete vector-valued GFF, including its covariance structure and range. Using its convergence to the continuum field under appropriate scaling, we also derive results about the range of the continuum GFF in various dimensions.

Motivated by connections to the low-temperature regime of the planar spin O(N) model, we then study the two-dimensional vector-valued Dirichlet GFF and its massive lattice counterpart, conditioned to avoid a ball at every site of a proper subdomain. We show that, under this conditioning, the norm of the massless field exhibits entropic repulsion, while its angular components freeze at all mesoscopic scales. In contrast, the norm of the massive field remains uniformly bounded as the system size grows. In the scalar massive case, we further establish a phase transition in the size of the avoided interval.

Finally, we use the GFF to rigorously characterize the large-N limit of the spin O(N) model and its infinite-volume behavior, and show how its limiting law differs from that of the spherical model despite their agreement in free energy. We also provide a probabilistic derivation of the infinite-volume limit of the spherical model across temperature regimes. To treat the model at criticality, we establish sharp bounds on the so-called zero-average Green's function.