  In this paper, we study the annealed ferromagnetic Ising model on the
configuration model. In an annealed system, we take the average on both sides
of the ratio {defining the Boltzmann-Gibbs measure of the Ising model}. In the
configuration model, the degrees are specified. Remarkably, when the degrees
are deterministic, the critical value of the annealed Ising model is the same
as that for the quenched Ising model. For independent and identically
distributed (i.i.d.) degrees, instead, the annealed critical value is strictly
smaller than that of the quenched Ising model. This identifies the degree
structure of the underlying graph as the main driver for the critical value.
Furthermore, in both contexts (deterministic or random degrees), we provide the
variational expression for the annealed pressure. Interestingly, our rigorous
results establish that only part of the heuristic conjectures in the physics
literature were correct.

Probability