Conformal dimension of CLE_κ carpets and loops

Determine whether a quasisymmetric homeomorphism can reduce the Hausdorff dimension of the CLE_κ carpet or of an individual CLE_κ loop, equivalently, determine the conformal dimension of the CLE_κ space and of a CLE_κ loop for κ in (8/3, 8], and ascertain whether these conformal dimensions equal their known Hausdorff dimensions 2 − ((3κ − 8)(8 − κ))/(32κ) for the CLE_κ space and 1 + κ/8 for a CLE_κ loop.

Background

The paper recalls that the CLE_κ carpet has almost sure Hausdorff dimension 2 − ((3κ − 8)(8 − κ))/(32κ) for κ in (8/3, 8], and that each CLE_κ loop has almost sure Hausdorff dimension 1 + κ/8, due to local absolute continuity with SLE_κ.

The authors ask whether quasisymmetric maps can lower these dimensions, i.e., what the conformal dimensions are. They express the expectation (heuristic) that the CLE_κ conformal dimension equals its Hausdorff dimension, possibly due to a product-like structure, but this is not established.

References

Question 2. Recall the Hausdorff dimension of $CLE_\kappa$, $\kappa \in (\frac{8}{3}, 8]$ space is a.s. $2-\frac{(3\kappa-8)(8-\kappa)}{32\kappa}$. Moreover, the Hausdorff dimension of one $CLE_\kappa$ loop is a.s. $1+\frac{\kappa}{8}$ since any loop in a $CLE_\kappa$ configuration is locally absolutely continuous with $SLE_\kappa$. Can one lower these by applying a quasisymmetric homeomorphism? We define the conformal dimension of a metric space be the infimum of the Hausdorff dimensions of all its quasisymmetric images.

— Quasisymmetric geometry of low-dimensional random spaces  (2412.06366 - Cai et al., 2024) in Section 6 (Further questions), Question 2