Diffusion Limited Aggregation is a very simple mathematical model which describes a wide range of natural phenomena. Despite its simplicity, there is very little progress in understanding its large-scale structure. Since its introduction by Witten and Sander over 40 years ago, there was only one mathematical result. In 1987 Kesten obtained an upper bound on the growth rate. In this talk I will discuss DLA and some related models and the recent progress in understanding DLA. In particular, a new simpler proof of Kesten result which generalizes to other aggregation models.

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.