We consider a random Hamiltonian $H:\Sigma\to\mathbb R$ defined on a compact
    space $\Sigma$ that admits a transitive action by a compact group $\mathcal G$.
    When the law of $H$ is $\mathcal G$-invariant, we show its expected free energy
    relative to the unique $\mathcal G$-invariant probability measure on $\Sigma$
    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 walk, 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 SK and SYK models.

    Source link


    Leave A Reply