  Lattice Gauge theories have been studied in the physics literature as
discrete approximations to quantum Yang-Mills theory for a long time. Primary
statistics of interest in these models are expectations of the so called
"Wilson loop variables". In this article we continue the program initiated by
Chatterjee (2015) to understand Wilson loop expectations in Lattice Gauge
theories in a certain limit through gauge-string duality. The objective in this
paper is to better understand the underlying combinatorics in the strong
coupling regime, by giving a more geometric picture of string trajectories
involving correspondence to objects such as decorated trees and non-crossing
partitions. Using connections with Free Probability theory, we provide an
elaborate description of loop expectations in the planar setting, which
provides certain insights about structures of higher dimensional trajectories
as well. Exploiting this, we construct an example showing that in any
dimension, the Wilson loop area law lower bound does not hold in full
generality.
