Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 209 / 4 / a / ii

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- Solution ii

Solution

Ferromagnetic monotonicity makes the plus-boundary expectation decrease as the finite volume grows, so

⟨ σ_{0} ⟩_{β, 0}^{Z^{2}, +} = lim_{L \to \infty} ⟨ σ_{0} ⟩_{Λ_{L}, β, 0}^{+}

exists. The uniform lower bound from part i passes to the limit, proving positive spontaneous magnetization at sufficiently large

β

.

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