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Supersymmetric circular Wilson loops in $\mathcal{N}=4$ Super-Yang-Millstheory are discussed starting from their Gaussian matrix model representations.Previous results on the generating functions of Wilson loops are reviewed andextended to the more general case of two different loop contours, which isnecessary to discuss coincident loops with opposite orientations. Acombinatorial formula representing the connected correlators of multiply woundWilson loops in terms of the matrix model solution is derived. Two new resultsare obtained on the expectation value of the circular Wilson loop, theexpansion of which into a series in $1/N$ and to all orders in the 't~Hooftcoupling $\lambda$ was derived by Drukker and Gross about twenty years ago. Theconnected correlators of two multiply wound Wilson loops with arbitrary windingnumbers are calculated as a series in $1/N$. The coefficient functions arederived not only as power series in $\lambda$, but also to all orders in$\lambda$ by expressing them in terms of the coefficients of the Drukker andGross series. This provides an efficient way to calculate the $1/N$ series,which can probably be generalized to higher-point correlators.

Exact 1/N expansion of Wilson loop correlators in $\mathcal{N}=4$ Super-Yang-Mills theory