Vortex condensate and critical exponents in the $(2+1)$-dimensional $\mathrm{O}(2)$ model

The vortex in the $(2+1)$-dimensional $\mathrm{O}(2)$ model is studied via numerical simulations in a fully non-perturbative lattice regularization. We compute the vortex condensate and susceptibility to determine its critical exponents and a renormalized condensate in the continuum limit. Together with recent results on the vortex mass, this gives a complete picture of the scaling behaviour of the vortex operator in this model and sheds light on the statistical mechanics of topological excitations.