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

Table of contents
- Solution b5

Solution

A connected set of at least

m

triangular-lattice sites has diameter at least

c_{0} m

. Part (b4) gives

P (0 lies in a finite cluster with at least m sites) \leq C e^{- c c_{0} m} \leq e^{- v 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 m}

. The upper and lower bounds therefore identify the correct stretched-exponential exponent

m

.

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