Establish conditional convergence of CLE16/3 percolation given CLE3

Show that the law of the colored CLE16/3 configuration conditionally on the nested CLE3 is the scaling limit of the corresponding discrete conditional laws of the FK-Ising configurations given the discrete Ising-interface configurations.

Background

A natural alternative reconstruction strategy samples nested CLE3, conditionally samples the continuum FK-Ising clusters through CLE percolation, and then reconstructs the magnetization field from the colored CLE16/3. To justify that this procedure produces the same field as the discrete Edwards–Sokal construction, one needs convergence of the conditional law of the FK-Ising scaling limit given the Ising interfaces.

The paper explains that existing continuum CLE-percolation results describe the coupling and show that the CLE3 can be reconstructed from the colored CLE16/3, but do not establish the required conditional-law convergence. Proving this statement would close the gap in that approach.

References

A natural strategy to actually prove the measurability of the IMF would therefore be to prove a strengthening of Theorem \ref{th. convergence ising g\xE9om\xE9trique + fk perco lehmkueller} (or perhaps better said, the "other direction" of the result described in the remark following that theorem): Show that the law of $\omega$ conditionally on $\Gamma$ is the limit in law of the conditional law of the $\omegaa$ w.r.t. $\Gammaa$, i.e. a "continuous independent percolation" inside $\Gamma$.

— The critical Ising magnetization field can be reconstructed from its $+/-$ interfaces  (2610.02064 - Cahen et al., 1 Oct 2026) in Section "Insights and limits of an approach via CLE percolation," Open Problem (label op la loi du cle(16/3) conditionnellement au cle(3) vient du discret)