A connected set of at least $m$ triangular-lattice sites has diameter at least $c_0\sqrt m$. Part (b4) gives
$$
\mathbb P(0\text{ lies in a finite cluster with at least }m\text{ sites})
\leq Ce^{-c c_0\sqrt m}
\leq e^{-v\sqrt m}
$$
after 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 $\sqrt m$.


Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-209/2/b5/solution