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Math-2884 · mathematics · Probability

BKT scaling and the endpoint stiffness for finite-range integer heights

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For each admissible finite-range symmetric integer-height interaction J in the stated planar class, is the roughening threshold finite and positive, with delocalization including equality? At every $\beta \geq b_c(J)$, ask for full-sequence torus-field convergence to $\operatorname{sqrt}(\beta _eff/v_J^2)$ times a GFF in $H^{-3}$, the matching Laplace functional, $\beta _eff(b_c)=8\pi $ $v_J^2$, an infinite right derivative at $b_c$, and $\beta _eff=\beta +O(\exp (-c_J$ beta)) at high beta. For square spread-out $J_\rho $ also ask $b_c(J_\rho )/(8\pi $ $v_{J_\rho }^2)\to 1$.

Short English statement

For each admissible finite-range symmetric integer-height interaction J in the stated planar class, is the roughening threshold finite and positive, with delocalization including equality? At every $\beta \geq b_c(J)$, ask for full-sequence torus-field convergence to $\operatorname{sqrt}(\beta _eff/v_J^2)$ times a GFF in $H^{-3}$, the matching Laplace functional, $\beta _eff(b_c)=8\pi $ $v_J^2$, an infinite right derivative at $b_c$, and $\beta _eff=\beta +O(\exp (-c_J$ beta)) at high beta. For square spread-out $J_\rho $ also ask $b_c(J_\rho )/(8\pi $ $v_{J_\rho }^2)\to 1$.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2884-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2884

The height theorem supplies each requested assertion for the admissible interaction $J$: the positive finite threshold and roughness at equality, the field and Laplace-functional limits, the endpoint value $\beta_{\rm eff}(b_c)=8\pi v_J^2$, the infinite right slope, the high-temperature estimate and the spread-out asymptotic. The normalized endpoint is $8\pi$, while the unnormalized nearest-neighbor value is $2\pi$ because $v_J^2=1/4$. The full label concerns these height-model clauses; the separate spin-field theorem is an additional result.

BKT scaling and the endpoint stiffness for finite-range integer heights

OpenAI's claim

OpenAI claims that finite-range integer heights have a Gaussian-free-field scaling limit at all rough-phase parameters and also at the phase boundary. In the parameter convention used here, the endpoint effective temperature is $8\pi v_J^2$, so its normalized ratio to $v_J^2$ is $8\pi$. The paper also gives spin-field limits for the Villain and XY models in its specified low-temperature regime.

Mathematical statement of the claimed result

Result 1.
Let $J\subset\mathbb Z^2\setminus\{0\}$ be finite, square-symmetric, and contain nearest neighbors. Define $\Delta_Ju(x)=|J|^{-1}\sum_{y\in J}(u(x+y)-u(x))$ and $v_J^2=(2|J|)^{-1}\sum_{y\in J}y_1^2$. For heights in $2\pi\mathbb Z$ with Gaussian weight $\exp[-(\sigma,-\Delta_J\sigma)/(2\beta)]$, the zero-boundary second-moment threshold $\beta_c$ is claimed finite and positive, with bounded variance below it and unbounded variance at/above it. On tori of side $L^N$, the centered distribution $H_N=(L^N)^{-2}\sum_x(\sigma_x-\bar\sigma)\delta_{x/L^N}$ satisfies \[ H_N\Longrightarrow\sqrt{\beta_{\mathrm{eff}}(J,\beta)/v_J^2}\,\Phi_{\mathbb T^2} \quad\text{in }H^{-3}\qquad(\beta\ge\beta_c). \] For mean-zero smooth $f$, log Laplace transforms tend to $\beta_{\mathrm{eff}}(f,(-\Delta)^{-1}f)/(2v_J^2)$. The asserted endpoint and asymptotics are \[ \beta_{\mathrm{eff}}(J,\beta_c)=8\pi v_J^2,\quad \lim_{\beta\downarrow\beta_c}\frac{\beta_{\mathrm{eff}}(J,\beta)-\beta_{\mathrm{eff}}(J,\beta_c)}{\beta-\beta_c}=+\infty, \quad\beta_{\mathrm{eff}}=\beta+O_J(e^{-c_J\beta}). \] The effective parameter is differentiable above threshold; for square interaction ranges $J_\rho$, $\beta_c(J_\rho)/(8\pi v_{J_\rho}^2)\to1$. Nearest-neighbor $v_J^2=1/4$ gives critical effective temperature $2\pi$. For each interaction range $J$, the torus side base is selected as an integer $L=L(J)\ge2$. With zero-boundary expectation in growing boxes of radius $R$, the threshold convention is \[ \beta_c=\inf\{\beta>0:\sup_R\mathbb E_{R,\beta}^{0}\sigma_0^2=\infty\}. \]

Outline of the argument

The height analysis controls renormalized covariance and the universal endpoint stiffness across finite square-symmetric interaction sets. Spin--height duality and Green-function normalization then identify the corresponding spin observables as imaginary exponentials of the Dirichlet Gaussian field.

Formal verification

Family 216 has no Lean scope page in this release. Its selected result is not presented as formalized. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

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