Wilson loops in large symmetric representations through a double-scaling limit

We derive exact formulas for circular Wilson loops in the N = 4 and N = 2* theories with gauge groups U(N) and SU(N) in the k-fold symmetrized product representation. The formulas apply in the limit of large k and small Yang-Mills coupling g, with fixed effective coupling kappa g(2)k, and for any finite N. In the SU(2) and U(2) cases, closed analytic formulas are obtained for any k, while the 1/k series expansions are asymptotic. In the N >> 1 limit, with N << k, there is an overlapping regime where the formulas can be confronted with results from holography. Simple formulas for correlation functions between the k-symmetric Wilson loops and chiral primary operators are also given.