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Math-2342 · mathematics · Probability

Variational Principles for Dilute Spin Systems

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
For dilute ferromagnets and sparse spin glasses, does the GREM-associated trial order parameter give the correct variational formula for the pressure? Does the numerator have a direct differential-equation formulation analogous to the Parisi equation? For the dilute ferromagnet, can one prove $\log Z_{N+M}\leq\log Z_N+\log Z_M$ directly? How should the pressure variational principle be formulated to exhibit appropriate trial order parameters, an entropy-minus-internal-energy interpretation, or a Legendre structure?

Formulation references: [S1], [S2].

Short English statement

For dilute ferromagnets and sparse spin glasses, does the GREM-associated trial order parameter give the correct variational formula for the pressure? Does the numerator have a direct differential-equation formulation analogous to the Parisi equation? For the dilute ferromagnet, can one prove $\log Z_{N+M}\leq\log Z_N+\log Z_M$ directly? How should the pressure variational principle be formulated to exhibit appropriate trial order parameters, an entropy-minus-internal-energy interpretation, or a Legendre structure?

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Statement licence: CC-BY-SA-4.0 · Source file

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Math-2342-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2342

The main theorem identifies the pressure infimum over finite hierarchical cavity trial laws for the even-arity Panchenko–Talagrand factorized integrable positive dilute Ising class. This answers the corresponding sparse-spin-glass variational clause, but not every model or the bundled differential-equation, direct ferromagnetic subadditivity, and Legendre-structure questions. This entry records partial support; complete coverage of the question is not established by this catalogue.

Variational Principles for Dilute Spin Systems

OpenAI's claim

OpenAI presents a mathematical result relevant to this question in the selected manuscript. For diluted even-arity Ising interactions in the Panchenko--Talagrand class, the limiting pressure is claimed to equal the infimum of finite hierarchical cavity trial values under first-moment integrability.

Mathematical statement of the claimed result

Result 1.
Fix even $p\ge2$, $\alpha>0$, independent $M_N\sim\operatorname{Pois}(\alpha N)$, interaction copies $\theta_k$, field copies $h_i$, and uniformly sampled vertex indices with replacement. Set \[ -H_N(\sigma)=\sum_{k=1}^{M_N}\theta_k(\sigma_{i_{1,k}},\ldots,\sigma_{i_{p,k}})+\sum_i h_i\sigma_i, \quad F_N=N^{-1}\mathbb E\log\sum_{\sigma\in\{\pm1\}^N}e^{-H_N(\sigma)}. \] Assume $\mathbb E\max_s|\theta(s)|+\mathbb E|h|<\infty$, and a factorization \[ e^{\theta(s)}=a\left(1+b\prod_{\ell=1}^pf_\ell(s_\ell)\right),\quad a>0,\quad|b\prod_\ell f_\ell(s_\ell)|<1,\quad \mathbb E(-b)^n\in[0,\infty)\quad(n\ge1), \] where the $f_\ell$ are iid random functions and independent of $b$. The claimed exact pressure is \[ \lim_{N\to\infty}F_N=\inf_{r\ge0}\Phi_r. \] To define $\Phi_r$, put $q_x(s)=e^{xs}/(2\cosh x)$ and \[ C_\theta(x_1,\ldots,x_p)=\sum_{s\in\{\pm1\}^p}e^{\theta(s)}\prod_{\ell=1}^pq_{x_\ell}(s_\ell),\quad U_\theta(x_1,\ldots,x_{p-1};\epsilon)=\log\sum_{s\in\{\pm1\}^{p-1}}e^{\theta(s,\epsilon)}\prod_{\ell<p}q_{x_\ell}(s_\ell). \] Let $\mathcal M_1=\mathcal P(\mathbb R)$ and $\mathcal M_{j+1}=\mathcal P(\mathcal M_j)$, with $\mathcal P$ denoting Borel probability laws. For depth $r\ge1$, choose $\zeta\in\mathcal M_{r+1}$ and $0<m_1<\cdots<m_r<1$. Every independent message label follows $\eta_0\sim\zeta$, $\eta_{d+1}\mid\eta_d\sim\eta_d$ for $0\le d\le r-2$, then $x\mid\eta_{r-1}\sim\eta_{r-1}$. These variables are independent of physical disorder. With independent $k\sim\operatorname{Pois}(\alpha p)$, copies $\theta_j$ and field $h$, set \[ V_{\mathrm s}=\tfrac12\sum_{\epsilon=\pm1}\exp\left(h\epsilon+\sum_{j=1}^kU_{\theta_j}(x_1^j,\ldots,x_{p-1}^j;\epsilon)\right),\quad V_{\mathrm e}=C_\theta(x_1,\ldots,x_p). \] Define $\mathcal F_d$ to contain physical disorder and all message ancestors through depth $d$. Starting at $T_rV=V$, recurse \[ T_dV=\left(\mathbb E[(T_{d+1}V)^{m_{d+1}}\mid\mathcal F_d]\right)^{1/m_{d+1}},\quad \Phi_r=\inf_{\zeta,\boldsymbol m}\left(\log2+\mathbb E\log T_0V_{\mathrm s}-\alpha(p-1)\mathbb E\log T_0V_{\mathrm e}\right). \] At depth zero take iid real messages with law $\zeta$ and omit the power means. Interaction and field laws remain fixed as $N$ varies. Distinct interactions, field samples, Poisson counts and sampled indices are independent; message samples are also independent of this physical disorder.

Outline of the argument

Finite auxiliary replica trees and bounded perturbations concentrate all finite multioverlaps on average over branching depths. A delayed-branching induction controls the cavity expansion before messages are redrawn independently; system size is sent to infinity before hierarchy depth.

Formal verification

For family 221, the documented theorem identifies pressure with the finite-hierarchy cavity infimum for the specified even-arity class and its factorization, positivity, and integrability assumptions. The listed Comparator files are DilutedSpin.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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