Factor models on locally tree-like graphs

Understand

We consider homogeneous factor models on uniformly sparse graph sequences converging locally to a (unimodular) random tree $T$, and study the existence of the free energy density $\phi$, the limit of the log-partition function divided by the number of vertices $n$ as $n$ tends to infinity.

• We provide a new interpolation scheme and use it to prove existence of, and to explicitly compute, the quantity $\phi$ subject to uniqueness of a relevant Gibbs measure for the factor model on $T$.
• By way of example we compute $\phi$ for the independent set (or hard-core) model at low fugacity, for the ferromagnetic Ising model at all parameter values, and for the ferromagnetic Potts model with both weak enough and strong enough interactions.
• Even beyond uniqueness regimes our interpolation provides useful explicit bounds on $\phi$.