Conformal invariance of random currents: a stability result

We show the convergence of the single sourceless critical random current to a limit identifiable with the nested CLE(3). Our approach is based on viewing the random current as a perturbation of the Ising interface, which is known to converge to CLE(3). Instead of focusing solely on the random current, we provide a general framework for the stability of scaling limits under the perturbation by superimposing an independent Bernoulli percolation.