Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 209 / 2 / b5 / Solution

A connected set of at least $m$ triangular-lattice sites has diameter at least $c_0\sqrt{m}$. Part (b4) givesafter reducing $v$ to absorb small $m$. Since $\#H_N$ is of order $N^2$, the lower bound in part (a4) is also of order $e^{-C\sqrt{m}}$. The upper and lower bounds therefore identify the correct stretched-exponential exponent $\sqrt{m}$.