Exponential growth rate of the critical set size for the configuration model Ising dynamics

Determine the existence of a deterministic constant c⋆ ∈ (0,1) such that N^{-1}log|C⋆_N| converges in probability to c⋆ as N→∞, where C⋆_N is the set of critical configurations (min–max states along optimal crossover paths between all-minus and all-plus) for the Stochastic Ising Model with Glauber dynamics on the configuration model.

Background

In metastability, the critical set C⋆_N (saddle-configurations realizing the min–max barrier) controls the transition geometry. For configuration model graphs, C⋆_N is random and its cardinality is believed to grow exponentially in N, but the precise rate is unknown.

Conjecture 2.10 proposes a deterministic limit for N^{-1}log|C⋆_N|. Establishing this would elucidate the geometric complexity of critical gates in random sparse environments.