Ising universality under weak interactions and random bonds

Source and verification

This article explains family 218 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

Overview

Weak changes in a lattice interaction can alter microscopic amplitudes while preserving a continuum interface law. Family 218 treats deterministic and random perturbations of planar Ising models. Its six companions separate bulk fields, spin interfaces, disorder fluctuations, and the finite conditional comparisons used to connect them.

🏷️ Preliminaries: Ising interfaces and two kinds of averaging

An Ising configuration assigns ${\sigma }_x\in \{-1,1\}$ to each vertex. A nearest-neighbor ferromagnetic weight is proportional to $\exp (\beta {\sum }_e{J}_e{\sigma }_x{\sigma }_y)$. With plus spins on one boundary arc and minus spins on the other, a distinguished interface separates the two boundary-connected spin regions.

Chordal ${\mathrm{SLE}}_{\kappa }$ is a conformally invariant random curve described in the upper half-plane by Loewner evolution:

$$\begin{array}{c}{\partial }_t{g}_t(z)=\frac{2}{g_t(z)-W_t},\\ W_t=\sqrt{\kappa }{B}_t.\end{array}$$

Time here is half-plane capacity; curve convergence can instead forget parametrization while retaining traversal order. Critical Ising spin interfaces correspond to $\kappa =3$. FK–Ising cluster boundaries are different curves, treated in family 223.

For random bonds, first sample an environment $\omega =(J_e)$ and then sample spins. A quenched law conditions on $\omega$; an annealed law averages it out. Convergence of the annealed interface law does not imply convergence of its conditional law for a typical environment.

🏷️ A map of the six companions

The bulk-field paper considers weak, finite-range, translation- and square-symmetric even-spin interactions, allowing signed coefficients. It claims universality of mixed bulk spin and energy correlations along a tuned critical branch, with observable normalizations. The energy must be centered in the actual perturbed state. (OpenAI, 2026) The deterministic interface paper treats weak finite-range contour interactions and claims ${\mathrm{SLE}}_3$ limits with its stated domain and exterior-condition hypotheses. (OpenAI, 2026b)

For bonds $J_e=1+\epsilon {\xi }_e$ with iid bounded centered nondegenerate ${\xi }_e$ and sufficiently small fixed $\epsilon$, the general disorder companion claims that the quenched critical spin-interface law converges to ${\mathrm{SLE}}_3$ in probability over the environment. The critical temperature is identified with the physical magnetization threshold. (OpenAI, 2026b) A separate companion treats symmetric two-valued disorder. (OpenAI, 2026c)

The logarithmic-fluctuation companion, assuming its deterministic critical reference estimates, claims a relative second-moment growth of order $(\log r{)}^{1/4+o(1)}$ for the specified spin correlation. (OpenAI, 2026b) The finite-comparison companion supplies a buffered comparison for Ising conditional laws and develops stopping-band applications. (OpenAI, 2026a)

These statements concern different observables and hypotheses. In particular, convergence of the random interface law does not require all microscopic correlation amplitudes to concentrate.

🏷️ A precise interpretation of quenched convergence

Let ${\mathcal{L}}_{\omega }({\gamma }_{\delta })$ be the conditional interface law and $\mathcal{L}(\gamma )$ the deterministic SLE law. For a bounded metric $d_{\mathrm{BL}}$ testing bounded Lipschitz functions, the conclusion has the form

$$d_{\mathrm{BL}}({\mathcal{L}}_{\omega }({\gamma }_{\delta }),\mathcal{L}(\gamma ))\underset{\delta \downarrow 0}{\overset{{\mathbb{P}}_{\omega }}{\to }}0.$$

Because the distance is bounded, this also gives convergence of its mean: split the expectation into distances at most $\eta$ and greater than $\eta$, then let $\delta \to 0$ and $\eta \to 0$. Thus the annealed limit follows. The reverse implication fails: two different conditional laws can have the same average. Nor does convergence in probability imply almost-sure convergence along the full mesh sequence.

🏷️ The proposed mechanism: local disorder and buffered interfaces

The random-bond argument represents the model at larger scales by random local perturbations of pure critical Ising. The thermal coordinate is tuned away. A disorder-variance coordinate has a negative quadratic correction and decays roughly as the reciprocal of the scale index, hence only logarithmically in physical length.

Such slow decay would be difficult to sum without geometry. A local change affects a distant crossing only when several paths traverse the intervening annulus. The corresponding passage estimates provide an extra power of the scale ratio, making the crossing errors summable. Large local disorder values are handled through environmental moments and local integrations, rather than a uniform bound over every realization.

For nonsymmetric bond laws, primal and dual scale trajectories initially select potentially different temperatures. Exact duality, torus winding probabilities, and a positive temperature-response estimate are used to show they agree. This is needed before identifying the physical critical point.

Finally, buffered conditional comparisons couple the perturbed interface to pure interfaces in nearby domains. The pure comparison marginal must stay correct after conditioning on the original environment; otherwise the result would only be annealed. A separate multiple-passage estimate retains traversal order, which a comparison of curve traces alone would not control.

The finite comparison underlying this step bounds a likelihood ratio by $\exp (\pm 4\mathrm{artanh}{q}_0)$, where $q_0$ is an appropriate FK connection probability after pinned vertices are deleted. It quantifies how a buffer screens distant conditions. It does not by itself prove the renormalization flow or the continuum curve limit.

🏷️ Key steps and proof outlines

A finite likelihood-ratio estimate from two effective spins

The buffered-comparison theorem starts with a finite ferromagnetic Ising graph, observed spins $I$, prescribed spins $J$, and common pinned spins $F$. Delete $F$ after retaining its contribution as external fields. Those fields can have either sign; oppositely signed pins are not wired together.

Let $p(a,b)>0$ be the resulting joint marginal on $I,J$, and set $f=\log p$. Ferromagnetic weights satisfy the lattice inequality

$$p(x\wedge y)p(x\vee y)\ge p(x)p(y).$$

The source proves that summing out a binary coordinate preserves this inequality. Applying it repeatedly makes the marginal log-supermodular. Its mixed log-odds rectangles obey

$$\begin{array}{rl}|f(a,b)+f(a^{\prime },b^{\prime })-f(a,b^{\prime })-f(a^{\prime },b)|& \le L,\\ L=f(+,+)+f(-,-)-f(+,-)-f(-,+).& \end{array}$$

The bound follows by writing the difference from the extreme rectangle as a sum of four nonnegative ordered rectangle differences. It applies even when $a,a^{\prime }$ or $b,b^{\prime }$ are incomparable.

Restrict each terminal set to a common sign. This contracts it to one effective spin and gives weights proportional to $e^{Kst+As+Bt}$, where $K=L/4\ge 0$. Add fields $-A,-B$ to balance the two means. Their covariance is then $\tanh K$. The imported arbitrary-field covariance inequality bounds it by the zero-field connection probability $q_0$ on the contracted, vertex-deleted graph. Thus $L\le 4\mathrm{artanh}{q}_0$.

For fixed $b,b^{\prime }$, the log density ratio on $I$ has oscillation at most $L$. The density ratio has mean one under its denominator law, so its range straddles one. Therefore it lies between $e^{-L}$ and $e^L$ pointwise. Averaging preserves the inequality for mixtures. Annular buffers make $q_0$ small, giving a conditional comparison that survives arbitrary common pins. (OpenAI, 2026a)

Why observable normalizations converge in the deterministic model

The bulk-field proof carries each spin or energy insertion as a marked derivative of the local scale transformation. After multiplying by its expected scale dimension, the unperturbed one-source map has the form

$$A_j=\left(\begin{array}{cc}1& B_j\\ 0& D_j\end{array}\right),\Vert B_j\Vert \le C,\Vert D_j\Vert \le \rho <1.$$

The first coordinate is the distinguished observable amplitude; the second contains decaying corrections. On the tuned interacting trajectory the map is $A_j+E_j$, with $\Vert E_j\Vert \le C|\lambda |{\gamma }^j$ for $\gamma <1$.

Products of the $A_j$ are uniformly bounded because the lower block contracts and the upper off-diagonal block is a geometric sum. Expanding a perturbed product at the locations of its $E_j$ factors bounds it by a constant times ${\prod }_j(1+C\Vert E_j\Vert )$, which converges. If the source coordinates are ${\alpha }_j,w_j$, then

$$\begin{array}{rl}\Vert w_j\Vert & \le {\rho }^j\Vert w_0\Vert +C|\lambda |\sum _{i<j}{\rho }^{j-1-i}{\gamma }^i,\\ |{\alpha }_{j+1}-{\alpha }_j|& \le C\Vert w_j\Vert +C|\lambda |{\gamma }^j.\end{array}$$

The stable remainder and amplitude increments are summable. Their limits define the common $Z_{\sigma }$ and $Z_{ϵ}$, independent of the domain and other marks. At zero perturbation both equal one, so continuity keeps them nonzero nearby. Energy insertions are first considered modulo constants; the final observable is centered in the actual interacting state. This prevents a surviving scalar term from overwhelming the normalized energy field. (OpenAI, 2026b)

Geometry compensates for logarithmically slow disorder decay

In the random-bond problem the variance coordinate decays only as $g_j\asymp 1/j$. Summing a bare error of this size over scales would diverge. Crossing comparisons instead charge a local change only when the changed square is pivotal for a macroscopic test.

The source distinguishes three passage bounds: four bulk passages have exponent $p_4>2$; two passages confined to a half-plane have exponent $p_h>1$; a single bulk passage has some exponent $p_1>0$. At scale $r$ in a region of diameter $R$, there are order $(R/r{)}^2$ candidate bulk squares. Multiplying by the four-passage cost $(r/R{)}^{p_4}$ leaves the positive power $(r/R{)}^{p_4-2}$. Along a wall the count is order $R/r$, so the half-plane exponent similarly leaves $(r/R{)}^{p_h-1}$. These geometric gains are summable over separated scales.

The argument first chooses annular ratios with strict probability slack, then starts sufficiently far along the random trajectory. Environmental moment estimates control rare large local perturbations. The conditional passage comparison retains distinct marks for repeated traversals; it is not an independence assertion about those traversals. This is how a slowly vanishing random perturbation can still give the pure law for macroscopic tests. (OpenAI, 2026e)

Matching primal and dual tuning without symmetric disorder

For general bounded centered bond disorder, primal and exact dual scale constructions initially select temperatures ${\beta }_{\mathrm{p}}$ and ${\beta }_{\mathrm{d}}$. The proof compares all four twisted partition functions on a torus. Local twists can be removed in contractible charts; because the interactions are even, their local centering factor is the same in all four sectors. After blocking, the residual total norm tends to zero in environmental probability, so the normalized four-component partition vector approaches the pure critical vector.

Exact duality in each fixed environment carries the dual vector back to the primal one. Thus the same primal winding-event probability $q_{N,\omega }(\beta )$ approaches the same nondegenerate pure value at both temperatures. No equality in distribution between primal and dual environments is asserted.

If ${\beta }_{-}<{\beta }_{+}$ were the two distinct temperatures, the fixed-environment differentiation estimate would give

$$q_{N,\omega }^{\prime }(\beta )\ge c_1{q}_{N,\omega }(\beta )(1-q_{N,\omega }(\beta )).$$

With high environmental probability, both endpoint values stay in $[c/2,1-c/2]$. Monotonicity keeps the whole interval there, and integration yields a positive difference at least $c_2({\beta }_{+}-{\beta }_{-})$. But both endpoints approach the same value. This contradiction identifies the two tunings; the criticality argument then identifies their common value with the physical threshold. (OpenAI, 2026d)

Relative correlation fluctuations and interface convergence coexist

The fluctuation companion transports a random spin amplitude ${\alpha }_j$ and proves its marked list is close to ${\alpha }_j$ times a deterministic source profile. Mixed source-background cumulants are retained in the recursion; replacing them by independent moments would change its coefficient. The neutral variance flow and marked moment recursion have coefficients whose averages, rather than necessarily their signs at every scale, determine the accumulated logarithm.

At terminal scale $J\asymp \log r$, its readout gives, writing $r_J$ for that spatial scale and $d$ for the spin dimension,

$$\begin{array}{rl}{r}_J^{2d}\mathbb{E}{C}_{\ast ,\omega }(y_1,y_2)& \asymp 1,\\ r_J^{4d}\mathbb{E}{C}_{\ast ,\omega }(y_1,y_2{)}^2& =J^{1/4+o(1)}.\end{array}$$

The error is controlled in $L^2$ by $O(J^{1/8-\eta +o(1)})$, smaller than the main $L^2$ norm $J^{1/8+o(1)}$. The triangle inequality and its reverse therefore preserve the leading second moment. Dividing by the squared mean gives the relative growth $(\log r{)}^{1/4+o(1)}$. This is a local amplitude fluctuation; the passage estimates above concern macroscopic geometry. (OpenAI, 2026c)

For interfaces, signed boundary barriers squeeze the target trace between pure comparison chords in displaced domains. The comparison must retain the correct pure marginal even conditional on the original environment to prove the quenched statement. Finally, a multiple-passage bound rules out reverse travel of positive diameter along a limiting chord. That last step upgrades agreement of traces to convergence of ordered curves. The deterministic companion uses the same separation between boundary barriers that identify the trace and passage estimates that control traversal order. (OpenAI, 2026f) (OpenAI, 2026d)

See Also

• Ising preliminaries supplies the basic Gibbs and boundary-condition vocabulary.
• Family 223 distinguishes FK–Ising curves from spin interfaces.
• Family 224 gives another comparison between quenched and annealed planar limits.

🏷️ Manuscripts and verification scope

• Quenched SLE₃ limits for general weak random-bond Ising models — pinned manuscript and companion source.
• Quenched SLE₃ Universality for the Weak Random-Bond Ising Model — pinned manuscript and companion source.
• Logarithmic Relative Fluctuations in the Weakly Disordered Planar Ising Model — pinned manuscript and companion source.
• Conformal universality of bulk Ising correlations under weak interactions — pinned manuscript and companion source.
• ${\mathrm{SLE}}_3$ universality for weak finite-range Ising interactions — pinned manuscript and companion source.
• Buffered comparison and stopping-band resolution in critical Ising — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The documented selection covers the finite buffered conditional comparison. It does not document the SLE limits, bulk universality, disorder flow, or the stopping-band approximation results. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026a. Buffered comparison and stopping-band resolution in critical Ising.

🐻  OpenAI 2026b. Conformal universality of bulk Ising correlations under weak interactions.

🐻  OpenAI 2026c. Logarithmic Relative Fluctuations in the Weakly Disordered Planar Ising Model.

🐻  OpenAI 2026d. Quenched SLE₃ limits for general weak random-bond Ising models.

🐻  OpenAI 2026e. Quenched SLE₃ Universality for the Weak Random-Bond Ising Model.

🐻  OpenAI 2026f. SLE₃ universality for weak finite-range Ising interactions.