gpt 6 astra pro attempt on Phys-12

Polyakov Conjecture: Absence of a Phase Transition for Planar O(n) Models with n >= 3

Definitions and mathematical statement

Place spins $\sigma_x\in S^{n-1}\subset{\mathbb{R}}^n$ at $x\in{\mathbb{Z}}^2$. The nearest-neighbor ferromagnetic Gibbs weight in a finite region is proportional to $\exp(\beta\sum_{\langle x,y\rangle}\sigma_x\cdot\sigma_y)$ relative to product uniform spherical measure. Two-point correlation is $\langle\sigma_0\cdot\sigma_x\rangle_\beta$ in the infinite-volume Gibbs convention of the source. For each $n\ge3$ and finite $\beta>0$, the target is exponential decay: \[ |\langle\sigma_0\cdot\sigma_x\rangle_\beta|\le C(n,\beta)e^{-c(n,\beta)|x|}, \qquad c(n,\beta)>0. \] The constants need not be uniform as temperature tends to zero. Absence of spontaneous magnetization alone is weaker than the claimed exponential bound.

Formulation and concept references: [S1].

Short English statement

Do planar O(n) spin correlations decay exponentially at every positive temperature once the spin dimension is at least three?

Research attempt

Current frontier.

Absence of spontaneous magnetization in two dimensions is weaker than a mass gap. Aru--Garban--Sepúlveda formulate the $n\ge3$ exponential-decay problem and study a percolative mechanism related to vector-valued Gaussian free fields; their results must not be read as a proof of exponential decay for every finite inverse temperature [S1], Section 1.1. We first reconstruct a high-temperature contraction, then isolate exactly what a scale-dependent improvement would need. The distinction from the planar $O(2)$ model is essential: that model has a rigorously established low-temperature nonexponential regime [E1].

Attack route.

Single-site mixing can give an explicit high-temperature criterion. A coarse-grained version could in principle extend it to all finite $\beta$, but only if an $n$-dependent mechanism makes a block-to-block influence kernel strictly subcritical. Alternatively one could prove a suitable finite-volume susceptibility criterion directly. We investigate the contraction route; neither a percolation analogy nor absence of magnetization is substituted for the missing contraction estimate.

Attempt.

At a site with neighbor sum $h$, the conditional spin distribution is \[ d\nu_h(s)=Z(h)^{-1}e^{\beta s\cdot h}\,d\omega(s),\qquad s\in S^{n-1}. \] When one neighboring spin changes from $u$ to $v$, the oscillation of the unnormalized log likelihood ratio is at most \[ \sup_s\beta s\cdot(u-v)-\inf_s\beta s\cdot(u-v) =2\beta|u-v|\le4\beta. \] If two probability densities have log likelihood ratio of oscillation at most $L$, their total variation distance is at most $\tanh(L/4)$. Here is a proof of the constant: for $r=dP/dQ$ with $a\le r\le b$ and $b/a\le e^L$, ${\mathbb{E}}_Qr=1$ gives \[ \tfrac12{\mathbb{E}}_Q|r-1|\le\frac{(1-a)(b-1)}{b-a}. \] Maximizing over $a\le1\le b$, $b/a\le e^L$, gives the symmetric extremum $a=e^{-L/2}$, $b=e^{L/2}$ and the asserted bound. Thus each of the four neighbors has influence at most $t=\tanh\beta$.

For completeness, the resulting comparison has a probabilistic interpretation. Couple two finite-volume heat-bath systems by maximal couplings. A disagreement at $x$ is propagated to a neighbor with coefficient at most $t$, so iteration gives a sum over nearest-neighbor walks. If \begin{equation}\label{wsref-0-65-71-31-38-37-3a-73-6d-61-6c-6c-62-65-74-61} q=4\tanh\beta<1, \end{equation} that sum converges and the influence between sites at graph distance $r$ is at most \[ \sum_{j\ge r}4^jt^j=\frac{q^r}{1-q}. \] The Dobrushin comparison theorem, whose proof uses these coupled finite-volume inequalities, makes this argument rigorous [E2]. One solves the finite-volume disagreement inequalities $d\le b+Cd$, with $C_{xy}=t$ on neighboring sites, and using $(I-C)^{-1}=\sum_{j\ge0}C^j$. Exhausting the lattice makes boundary influences tend to zero and yields uniqueness. Rotation invariance then gives $\langle\sigma_x\rangle=0$.

Conditioning on the spin at the origin and applying the comparison to each bounded coordinate function gives the conservative bound \begin{equation}\label{wsref-0-65-71-31-38-37-3a-68-69-67-68-54} |\langle\sigma_0\cdot\sigma_x\rangle| \le \frac{2n}{1-q}q^{|x|_1}\qquad(x\ne0). \end{equation} Indeed, the conditional expectation of each coordinate at $x$ changes by at most twice the influence bound as the spin at zero varies; its unconditional mean is zero. Multiply by the corresponding coordinate at zero, bounded by one, and sum over the $n$ coordinates. The factor $2n$ is not optimized. This recovers a high-temperature special case for $\beta<\operatorname{arctanh}(1/4)$, not the full assertion.

Finite-scale lemma. Let $g:{\mathbb{Z}}^2\to[0,1]$, and suppose for some integer $L\ge1$ and nonnegative kernel $K$ supported on $0<|z|_1\le L$ that \begin{equation}\label{wsref-0-65-71-31-38-37-3a-62-6c-6f-63-6b} g(x)\le\sum_zK(z)g(x-z)\quad(|x|_1>L),\qquad \sum_zK(z)=q_L<1. \end{equation} Then, with $m=\max\{0,\lfloor |x|_1/L\rfloor-1\}$, \[ g(x)\le q_L^m. \] To prove it, substitute the inequality $m$ times. Before the last substitution every reachable site still has norm greater than $L$, by the triangle inequality. The sum of all path weights is exactly $q_L^m$, while the terminal value is at most one. If $q_L=0$ the function vanishes outside the ball; otherwise this is exponential decay with rate $-L^{-1}\log q_L$, up to a harmless prefactor.

Applying this lemma to the absolute correlation would prove the stated target if, for every $n\ge3$ and finite $\beta$, one could construct an actual correlation comparison kernel satisfying (eq187:block). The lemma accommodates a correlation length growing arbitrarily rapidly with $\beta$; it does not demand an unphysical temperature-uniform mass gap.

Stress test.

The single-site estimate is independent of the distinction $n=2$ versus $n\ge3$. Therefore iterating it without new geometric information cannot explain the conjectured distinction. Worst-case block boundary influences may become larger, not smaller: replacing a block by a "renormalized spin" and assuming a smaller effective coupling is an unproved step. The finite-scale lemma is only a sufficient analytic criterion; no Simon--Lieb-type inequality with subcritical total weight at all temperatures has been proved here. Mermin--Wagner's conclusion does not provide that kernel, and slow decay to zero would not suffice. The low-temperature $O(2)$ results in [E1] actively falsify any dimension-insensitive argument purporting to establish the full all-temperature conclusion.

Outcome.

Outcome: Exploratory approach with unresolved gap. We proved an explicit high-temperature estimate and a finite-scale contraction lemma, both with transparent constants and quantifiers. They are elementary reconstructions, not a new mass-gap theorem. The missing step is an $n\ge3$-specific low-temperature estimate that produces a subcritical comparison kernel at some finite scale for every fixed $\beta$. No such estimate is established.

Sources

• S1 J. Aru, C. Garban, and A. Sepulveda, Percolation for 2D Classical Heisenberg Model and Exit Sets of Vector Valued GFF, Section 1.1, CMP 406 $2025$. https://link.springer.com/article/10.1007/s00220-024-05208-y. Catalogue formulation source.
• E1 Jürg Fröhlich and Thomas Spencer, The Kosterlitz--Thouless transition in two-dimensional abelian spin systems and the Coulomb gas, Communications in Mathematical Physics 81 $1981$, 527--602; low-temperature planar rotator correlations. https://doi.org/10.1007/BF01208273. Research reference; consulted 27 September 2026.
• E2 R. L. Dobrushin, The Description of a Random Field by Means of Conditional Probabilities and Conditions of Its Regularity, Theory of Probability and Its Applications 13 $1968$, no. 2, 197--224. Single-site influence contraction and comparison. https://doi.org/10.1137/1113026. Research reference; consulted 27 September 2026.