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At low energies or temperatures, maximally supersymmetric Yang-Mills theoryon $\mathbb R^{(t)}\times S^1$ with large $N$ gauge group $SU(N)$ and strongt'Hooft coupling is conjectured to be dual to the low energy dynamics of acollection of D0-branes on a circle. We construct thermal states in thegravitational side of the correspondence where we find a first-order phasetransition between states that are uniform on the $S^1$ and states that arelocalised on it. When compared with lattice computations that are nowavailable, these critical values provide the first instance where a first-orderphase transition is tested on both sides of gauge/gravity duality.

Localised and nonuniform thermal states of super-Yang-Mills on a circle