Unique giant component implies weak concentration conjecture

Let $(\mathbf V_n)$ be a sequence of finite edge-weighted graphs satisfying the background assumption (hypothesis (2)). Define the unique giant component property by

$$\sup_t \frac{C_n^{[2]}(t)}{|\mathbf V_n|}\to_p 0\quad\text{as }n\to\infty,$$

where $C_n^{[2]}(t)$ is the size of the second-largest component at time $t$. Let the weak concentration property be the property denoted by (w-c) in the source.

Unique giant component implies weak concentration conjecture. Under the background assumption, the unique giant component property implies the weak concentration property.

Equivalently, the limiting process $C_\infty(\cdot)$ may be deterministic and continuous or random and discontinuous, but cannot be random and continuous. This is presented as an open problem, and the source relates weak concentration to the inverse of the percolation function.