Energy barrier per vertex limit for Ising Glauber dynamics on the configuration model

Establish the existence of a deterministic constant γ⋆ such that N^{-1}Γ⋆_N converges in probability to γ⋆ as N→∞ for the Stochastic Ising Model with Glauber dynamics on the configuration model CM_N (with i.i.d. degrees having minimum at least 3 and finite mean), where Γ⋆_N denotes the energy barrier between the all-minus configuration and the all-plus configuration along the optimal crossover path.

Background

For the Ising model on sparse random graphs generated by the configuration model, metastable transitions between the all-minus and all-plus states are controlled by a random energy barrier Γ⋆_N and a prefactor K⋆_N. While upper and lower bounds on Γ⋆_N are known under certain regimes, a sharp characterization of its scaling in N is lacking.

Conjecture 2.9 posits a law of large numbers for the barrier: a deterministic per-vertex limit γ⋆. Proving this would significantly clarify metastable behavior in sparse disordered mean-field settings.