We present an inter-twining relation between operators on loop ensembles and connections.

We talk about the Hausdorff measure of level sets of the fields, say length of level lines of a planar field. Given two coupled stationary fields  $f_1, f_2$ , we estimate the difference of Hausdorff measure of level sets in expectation, in terms of $C^2$-fluctuations of the field $F=f_1-f_2$. The main idea in the proof is to represent difference in volume as an integral of mean curvature using the divergence theorem. This approach is different from using the Kac-Rice type formula as the main tool in the analysis.