  We study a distribution of thermal states given by random Hamiltonians with a
local structure. We show that the ensemble of thermal states monotonically
approaches the unitarily invariant ensemble with decreasing temperature if all
particles interact according to a single random interaction and achieves a
state $t$-design at temperature $O(1/\log(t))$. For the system where the random
interactions are local, we show that the ensemble achieves a state $1$-design.
We then provide numerical evidence indicating that the ensemble undergoes a
phase transition at finite temperature.
