We consider a random Hamiltonian H:Σ→R defined on a compact space Σ that admits a transitive action by a compact group G. When the law of H is G-invariant, we show its expected free energy relative to the unique G-invariant probability measure on Σ, which obeys a subadditivity property in the law of H itself. The bound is often tight for weak disorder and relates free energies at different temperatures when H is a Gaussian process. Many examples are discussed, including branching random walks, several spin glasses, random constraint satisfaction problems, and the random field Ising model. We also provide a generalization to quantum Hamiltonians with applications to the quantum Sherrington–Kirkpatrick and Sachdev–Ye–Kitaev models.
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