Surface sums for lattice Yang-Mills in the large-$N$ limit

We give a sum over weighted planar surfaces formula for Wilson loop expectations in the large-$N$ limit of strongly coupled lattice Yang-Mills theory, in any dimension. The weights of each surface are simple and expressed in terms of products of signed Catalan numbers. In establishing our results, the main novelty is to convert a recursive relation for Wilson loop expectations, known as the master loop equation, into a new peeling exploration of the planar surfaces. This exploration reveals hidden cancellations within the sums, enabling a deeper understanding of the structure of the planar surfaces. We view our results as a continuation of the program initiated in [CPS23] to understand Yang-Mills theories via surfaces and as a refinement of the string trajectories point-of-view developed in [Cha19a].