Macroscopic freezing of a vector Gaussian free field conditioned to avoid a ball

Source abstract

We consider a fixed finite number of independent two-dimensional discrete Gaussian free fields on a box of side $n$, with zero boundary values, conditioned so that the vector-valued field stays outside a fixed ball on a macroscopic interior domain. We prove that the field, divided by $\log n$, converges in distribution in $L^2$ to twice the equilibrium potential of the domain multiplied by a uniform random unit vector. We also identify the exact capacity cost of the conditioning event and obtain alignment on a deterministic density-one set of sites, with no restriction on their mutual distance. The proof combines the norm repulsion theorem of Korzhenkova and Sepúlveda with Gaussian large deviations and the equality case in the capacity variational problem.