Sherrington-Kirkpatrick模型中的临界自由能方差

中文摘要 AI 辅助

我们研究Sherrington-Kirkpatrick模型在临界点(β,h)=(1,0)处的归一化自由能F_N=N^{-1}log Z_N，并证明无条件方差界Var F_N(1)≤(1/(2N^2))log χ_SG(N)+O(N^{-2})，其中χ_SG(N)是有限尺寸自旋玻璃磁化率。任何磁化率界χ_SG(N)≤N^{γ+o(1)}因此给出方差系数γ/2：一个自包含的端点界和Talagrand的经典重叠界分别给出系数1/2和1/4，而Du和Huang最近证明的临界重叠标度χ_SG(N)≍N^{1/3}给出尖锐系数1/6。该界依赖于沿Ornstein-Uhlenbeck无序耦合的精确有限N恒等式，该恒等式将2N^2 Var F_N(1)表示为log χ_SG(N)减去一个非负积分余项，直至O_c(1)。该恒等式结合他们的方差公式和重叠标度，表明该余项有界，因此该界以等式成立：Var F_N(1)=(1/(2N^2))log χ_SG(N)+O(N^{-2})。对于每个固定的β<1，同一恒等式给出有界余项，并恢复尖锐的经典高温常数。

英文摘要

We study the normalized free energy $F_N=N^{-1}\log Z_N$ of the Sherrington-Kirkpatrick model at the critical point $(β,h)=(1,0)$, and prove the unconditional variance bound \[ \mathrm{Var}\,F_N(1)\le\frac{1}{2N^2}\logχ_{\mathrm{SG}}(N)+O(N^{-2}), \] where $χ_{\mathrm{SG}}(N)$ is the finite-size spin-glass susceptibility. Any susceptibility bound $χ_{\mathrm{SG}}(N)\le N^{γ+o(1)}$ thus yields the variance coefficient $γ/2$: a self-contained endpoint bound and Talagrand's classical overlap bound give the coefficients $1/2$ and $1/4$, while the critical overlap scaling $χ_{\mathrm{SG}}(N)\asymp N^{1/3}$ recently proved by Du and Huang gives the sharp coefficient $1/6$. The bound rests on an exact finite-$N$ identity along an Ornstein-Uhlenbeck disorder coupling, which expresses $2N^2\mathrm{Var}\,F_N(1)$ as $\logχ_{\mathrm{SG}}(N)$ minus a nonnegative integrated remainder, up to $O_c(1)$. The identity, combined with their variance formula and overlap scaling, shows that this remainder is bounded, so the bound holds with equality: $\mathrm{Var}\,F_N(1)=\frac{1}{2N^2}\logχ_{\mathrm{SG}}(N)+O(N^{-2})$. For every fixed $β<1$, the same identity gives a bounded remainder and recovers the sharp classical high-temperature constant.

发表机构

• Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（上海数学与交叉学科研究院）

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32 pages