  We study the Yang--Mills measure on the sphere with unitary structure group.
In the limit where the structure group has high dimension, we show that the
traces of loop holonomies converge in probability to a deterministic limit,
which is known as the master field on the sphere. The values of the master
field on simple loops are expressed in terms of the solution of a variational
problem. We show that, given its values on simple loops, the master field is
characterized on all loops of finite length by a system of differential
equations, known as the Makeenko--Migdal equations. We obtain a number of
further properties of the master field. On specializing to families of simple
loops, our results identify the high-dimensional limit, in non-commutative
distribution, of the Brownian loop in the group of unitary matrices.
