Limit of scaled prefactor times critical set size for the configuration model Ising dynamics

Establish the existence of a deterministic constant κ⋆ ∈ (1,∞) such that |C⋆_N| K⋆_N converges in probability to κ⋆ as N→∞, where K⋆_N is the metastable prefactor (the limit of e^{−βΓ⋆_N}E(τ) as β→∞) associated with the crossover from all-minus to all-plus in the Stochastic Ising Model on the configuration model.

Background

Beyond the exponential barrier, metastable transition times include a prefactor K⋆_N reflecting local dynamics near the critical set. In random graphs, both |C⋆_N| and K⋆_N are random and potentially correlated, making their product a meaningful aggregate quantity.

Conjecture 2.11 asserts convergence of |C⋆_N|K⋆_N to a deterministic limit κ⋆, which would sharpen the potential-theoretic description of metastability on sparse random graphs.