Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 215 / 2 / a

Table of contents
- Solution a

Solution

One step of the Glauber dynamics chooses a vertex

v

uniformly and resamples its spin from its conditional Ising model distribution given all other spins. If the proposed new spin is

x

and

S_{v} (σ) = \sum_{w \sim v} σ (w)

, its update probability is

\frac{1}{n} \frac{e ^{β x S_{v} (σ)}}{e ^{β S_{v} (σ)} + e ^{- β S_{v} (σ)}} = \frac{1 + t a n h { β x S _{v} ( σ )}}{2 n} .

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