Free Energy Subadditivity for Symmetric Random Hamiltonians

TL;DR

This paper proves a subadditivity property of expected free energy for symmetric random Hamiltonians, with applications to various classical and quantum models, revealing fundamental bounds and relations across different systems.

Contribution

It establishes a general subadditivity principle for free energies of symmetric random Hamiltonians, extending to quantum models and encompassing diverse examples like spin glasses and constraint satisfaction problems.

Findings

Expected free energy obeys subadditivity under symmetry.

Bounds are tight in weak disorder regimes.

Relations between free energies at different temperatures are derived.

Abstract

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.