  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.
