Vector-Valued Gaussian Free Field Conditioned to Avoid a Ball: Entropic Repulsion of the Norm and Freezing of Spins

We study the laws of the two-dimensional vector-valued Dirichlet Gaussian free field and its massive lattice counterpart, conditioned to avoid a ball at every site of a subdomain. We prove that, under this conditioning, the norm of the massless field exhibits entropic repulsion, while its angular components freeze at all mesoscopic scales. A key step in the analysis is showing that around any given point in the bulk of the range, the unconditioned field has no holes. In the massive case, the conditioned field behaves differently: its norm remains uniformly bounded as the system size grows, leading to the existence of infinite-volume Gibbs measures. Furthermore, in the scalar massive case, the system undergoes a phase transition in the size of the avoided interval: for small intervals, the system admits a unique infinite-volume limit, while for sufficiently large centered intervals, multiple such limits exist.