Large N phase transitions under scaling and their uses ∗

The eigenvalues of Wilson loop matrices in SU(N) gauge theories in dimensions 2,3,4 at infinite N are supported on a small arc on the unit circle centered at z = 1 for small loops, but expand to the entire unit circle for large loops. These two regimes are separated by a large N phase transition whose universal properties are the same in d = 2, 3 and 4. Hopefully, this large N universality could be exploited to bridge traditional perturbation theory calculations, valid for small loops, with effective string calculations for large loops. A concrete case of such a calculation would obtain analytically an estimate of the large N string tension in terms of the perturbative scale ΛSU(N).