Evidence for large N phase transition in Script N = 4 super Yang-Mills theory at finite temperature

The AdS/CFT correspondence provides valuable constraints on the possibleexact form of various physical quantities in the ${\cal N}=4$ super Yang-Millstheory in the large N limit. We examine the free energy as the expansions in asmall as well as in a large 't Hooft parameter $\lambda$. We argue that it isimpossible to smoothly extrapolate from the weak coupling regime to the strongcoupling regime, thus there must exist a large N phase transition in $\lambda$at a finite temperature. We also argue that there is no world-sheet instantonin the background of the Euclidean anti-de Sitter black hole.Comment: 12 pages, harvmac; v2: corrections to some formulas, one reference adde