Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 209 / 3 / 2

Table of contents
- Solution 2

Solution

Only the

2 d

edges incident to

x_{0}

depend on

Γ (x_{0})

. Conditional on the remaining field, write

m = \frac{1}{2 d} \sum_{y \sim x_{0}} Γ (y) .

Completing the square gives

\sum_{y \sim x_{0}} (Γ (x_{0}) - Γ (y))^{2} = 2 d (Γ (x_{0}) - m)^{2} + constant .

The conditional density is therefore proportional to

e^{- (u - m)^{2}}

, and hence

Γ (x_{0}) ∣ (Γ (x) : x \neq = x_{0}) \sim N (\frac{1}{2 d} \sum_{y \sim x_{0}} Γ (y), \frac{1}{2}) .

This is the Gibbs-Markov property of the discrete Gaussian free field in the normalization used by the paper.

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