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Math-2891 · mathematics · Probability

The all-temperature pressure of orthogonally invariant Ising disorder

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For $J_N=U_N$ $Lambda_N$ $U_N^T$ with $U_N$ Haar orthogonal, deterministic diagonal $Lambda_N$ whose empirical spectral laws approach compactly supported mu and whose extreme eigenvalues approach the edges of supp(mu), does the uniform-spin Ising pressure converge in mean and almost surely to $inf_p$ ${S(p)+G_1(p)}$, over the stated hierarchical trial measures? No independence of $U_N$ between different dimensions is required.

Short English statement

For $J_N=U_N$ $Lambda_N$ $U_N^T$ with $U_N$ Haar orthogonal, deterministic diagonal $Lambda_N$ whose empirical spectral laws approach compactly supported mu and whose extreme eigenvalues approach the edges of supp(mu), does the uniform-spin Ising pressure converge in mean and almost surely to $inf_p$ ${S(p)+G_1(p)}$, over the stated hierarchical trial measures? No independence of $U_N$ between different dimensions is required.

Sources

Statement licence: CC-BY-4.0 · Source file

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Math-2891-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2891

Haar-orthogonal Ising spin glasses without spectral outliers are claimed to have an all-temperature variational pressure formula, with admissible deterministic fields and a zero-temperature ground-energy limit. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

The all-temperature pressure of orthogonally invariant Ising disorder

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Haar-orthogonal Ising spin glasses without spectral outliers are claimed to have an all-temperature variational pressure formula, with admissible deterministic fields and a zero-temperature ground-energy limit.

Mathematical statement of the claimed result

Result 1.
Let $\Lambda_N$ be deterministic real diagonal matrices, with empirical eigenvalue measures tending weakly to a compactly supported probability law $\mu$ and extreme eigenvalues tending to the edges of its support. For Haar $U_N\in O(N)$, define \[ P_N=N^{-1}\log\left(2^{-N}\sum_{\sigma\in\{-1,1\}^N} e^{\sigma^{\mathsf T}U_N^{\mathsf T}\Lambda_NU_N\sigma/2}\right). \] Write $\lambda_+=\max\operatorname{supp}\mu$. For $x>0$, choose $b(x)>\lambda_+$ solving $\int(b(x)-\lambda)^{-1}\mu(d\lambda)=x$ when such a solution exists, and otherwise set $b(x)=\lambda_+$. Put $R(x)=b(x)-1/x$ and $R(0)=\int\lambda\,\mu(d\lambda)$. Let $\mathcal P$ consist of nondecreasing measurable $p:(0,1)\to[0,1]$, and set \[ D_p(r)=1-r-\int_0^1(p(s)-r)_+\,ds,\qquad G_1(p)=\frac12\int_0^1R(D_p(r))\,dr. \] The entropy term is defined as follows. For a finite nonnegative nondecreasing step field $h$, with values $h_i$ on $\zeta_i<u<\zeta_{i+1}$, $0=\zeta_0<\cdots<\zeta_k=1$, set $h_{-1}=0$ and \[ T_{a,v}f(z)=\begin{cases}a^{-1}\log\mathbb E e^{af(z+\sqrt vG)},&a>0,\\ \mathbb Ef(z+\sqrt vG),&a=0,\end{cases}\quad G\sim N(0,1). \] Starting with $F_{k-1}(z)=\log\cosh z$, recursively put $F_{i-1}=T_{\zeta_i,h_i-h_{i-1}}F_i$ for $i=k-1,\ldots,0$. Then \[ \ell(h)=F_{-1}(0)-h_{k-1}/2,\qquad S(p)=\sup_h\left(\ell(h)+\frac12\int_0^1p(u)h(u)\,du\right). \] At each fixed temperature, absorbed into $\Lambda_N$, the limiting pressure satisfies \[ \lim_N\mathbb EP_N=\lim_NP_N=\inf_{p\in\mathcal P}\{S(p)+G_1(p)\}, \] with the second limit almost sure, without requiring independence across $N$.

Outline of the argument

Finite spectral alphabets permit spectral-coordinate overlap identities and cascade calculus. Concavity of the Ising field functional and rotation identities give matching variational bounds; continuity extends the result to compact spectral laws and first-moment field limits, and temperature passage yields ground energy.

Formal verification

For family 234, the documented scope gives all-positive-temperature pressure, admissible external-field versions, and the zero-temperature value for the stated Haar-invariant compact-spectrum models. The listed Comparator files are InvariantIsing.lean. The inventory does not supply a named declaration for these Comparator entries. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

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