Polyakov's mass-generation conjecture: exponential decay at every temperature in the two-dimensional O(n) model, n at least 3

Theorem 1.1: for every integer $n\ge3$ and finite $\beta>0$ there are $A<\infty$, $m>0$ with $0\le\langle\sigma_x\cdot\sigma_y\rangle_{G,b}\le Ae^{-m\|x-y\|_2}$ for every finite subgraph $G$ of $\mathbb Z^2$ with free boundary and every coupling array $b\in[0,\beta]^E$; Corollary 1.2 passes this to free-boundary box limits, giving finite susceptibility. The companion 'Sharp mass bounds' proves, for $O(4)$, a positive gap of the full transfer operator (all local observables) in the unique periodic state at every $\beta>0$, and $m\asymp\sqrt\beta e^{-\pi\beta}$ for large $\beta$. Not shown: uniqueness of infinite-volume Gibbs states for general $n$, a full transfer gap for $n\ne4$ in this paper, or the low-temperature correlation length for general $n$.

Theorem 1.1: for every integer $n\ge3$ and finite $\beta>0$ there are $A<\infty$, $m>0$ with $0\le\langle\sigma_x\cdot\sigma_y\rangle_{G,b}\le Ae^{-m\|x-y\|_2}$ for every finite subgraph $G$ of $\mathbb Z^2$ with free boundary and every coupling array $b\in[0,\beta]^E$; Corollary 1.2 passes this to free-boundary box limits, giving finite susceptibility. The companion 'Sharp mass bounds' proves, for $O(4)$, a positive gap of the full transfer operator (all local observables) in the unique periodic state at every $\beta>0$, and $m\asymp\sqrt\beta e^{-\pi\beta}$ for large $\beta$. Not shown: uniqueness of infinite-volume Gibbs states for general $n$, a full transfer gap for $n\ne4$ in this paper, or the low-temperature correlation length for general $n$.