Expected-time asymptotics for binary border aggregation

Let

K$ be the depth parameter for the binary border aggregation model, let

\xi_K$ be the number of particles or the corresponding absorption time, and define

$$m_K=2^{K-\sqrt{2K}+\frac12\log_2K-\frac{1}{\log 2}+\frac{\log_2K}{4\sqrt{2K}}}.$$

Expected-time asymptotics conjecture. The expectation satisfies

$$\mathbb E\xi_K=m_K\bigl(1+O(K^{-1/2})\bigr)=2^{K-\sqrt{2K}+\frac12\log_2K-\frac{1}{\log 2}+\frac{\log_2K}{4\sqrt{2K}}+O(K^{-1/2})}.$$

The paper proves only the cruder estimate $\mathbb E\xi_K=(1+o(1))m_K$ and explains that stronger tail estimates would be needed to establish this conjecture; it is left open.