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We study the thermodynamics of the maximally supersymmetric Yang-Mills theorywith gauge group U(N) on R x S^3, dual to type IIB superstring theory on AdS_5x S^5. While both theories are well-known to exhibit Hagedorn behavior atinfinite N, we find evidence that this is replaced by Lee-Yang behavior atlarge but finite N: the zeros of the partition function condense into two arcsin the complex temperature plane that pinch the real axis at the temperature ofthe confinement-deconfinement transition. Concretely, we demonstrate this forthe free theory via exact calculations of the (unrefined and refined) partitionfunctions at N<=7 for the su(2) sector containing two complex scalars, as wellas at N<=5 for the su(2|3) sector containing 3 complex scalars and 2 fermions.In order to obtain these explicit results, we use a Molien-Weyl formula forarbitrary field content, utilizing the equivalence of the partition functionwith what is known to mathematicians as the Poincare series of trace algebrasof generic matrices. Via this Molien-Weyl formula, we also generate exactresults for larger sectors.

From Hagedorn to Lee-Yang: Partition functions of $\mathcal{N}=4$ SYM theory at finite $N$