Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 209 / 2

Let $\Delta$ be the torus graph Laplacian and let $v:{\Lambda }_L\to {\mathbb{R}}^n$ have zero spatial mean. Gaussian domination states, in a standard normalization, thatDifferentiating twice at zero gives the infrared bound: for every nonzero torus momentum $k$,up to the harmless normalization convention used for the Fourier transform and Hamiltonian.

The spin-length sum rule and Fourier inversion giveThe zero mode contains the square of the field-directed magnetization, while the infrared bound controls all nonzero modes. After $L\to \infty$ and then $h\downarrow 0$,Near zero, $\epsilon (k)\asymp |k{|}^2$, so the integral is finite exactly when $d\ge 3$. Choose ${\beta }_0$ larger than the resulting finite constant. Then for $\beta >{\beta }_0$ the right side is positive, and $m(\beta )\ge c>0$.

If $h\downarrow 0$ before $L\to \infty$, every finite torus retains global $O(n)$ symmetry and its magnetization is zero. The reversed iterated limit is therefore zero. The order of limits is what permits spontaneous magnetization.

The field leaves rotations fixing its unit direction $e$ as symmetries. Consequently the expectation vector is parallel to $e$:For any $u\in {\mathbb{R}}^n$, linearity therefore gives

Let $\vartheta$ be either permitted torus reflection and let ${\Lambda }_{+}$ be one reflected half. Under the product measure ${\mu }^{\otimes {\Lambda }_L}$, variables in the two open halves are independent and corresponding variables have the same law. For any $F$ depending on ${\Lambda }_{+}$,when the reflection has no fixed sites. If it fixes a layer of sites, condition on that layer; the same factorization gives a conditional square, whose expectation is nonnegative. This proves reflection positivity through sites. For an edge reflection the parity assumption makes the two halves pair exactly, so the first factorization applies there as well.