AI 中文总结
本文证明Sherrington-Kirkpatrick模型自由能上偏差指数为$\delta^{6/5}$，通过约束Parisi测度分析给出精确常数，并推广到更广模型及小场效应。
AI 中文摘要
设$Z_N$为逆温度$\beta>1$下Sherrington-Kirkpatrick模型的配分函数。复制理论（Kondor在临界温度附近，Parisi和Rizzo在所有温度下）预测，$N^{-1}\log Z_N$超过其典型值$\delta$的概率按$\exp(-Na\delta^{6/5})$衰减，其中常数$a$是显式的；Aronow和Lopatto最近证明了该阶的界。该指数是Parisi问题在零点携带原子$\theta$的约束下的Legendre变换。我们证明约束最小值超过平衡自由能$c_\beta\theta^5(1+o(1))$，其中$c_\beta=\frac{9}{640}\beta^3\rho_\beta(0)^{-3}$，$\rho_\beta(0)>0$是Parisi测度在零点的密度。利用该模型有效的分数矩公式，当$N\to\infty$然后$\delta\to0$时，得到指数$\frac56(6c_\beta)^{-1/5}\delta^{6/5}(1+o(1))$。约束极小化器具有边际首次正接触，首阶为$3\theta/(2\rho_\beta(0))$，带有原子$\theta/2$，在此接触点以下，在接触点处等于恒等式的函数与其相差一个首阶普适三次项。该平台微积分适用于每个满足$\xi'''(0)=0$且其Parisi测度在零点累积的混合模型；它还表明，一个小场$h$打开一个$|h|^{2/3}$阶的间隙，并向自由能添加一个$|h|^{10/3}$阶的项，并且Parisi测度在零点的原子（存在于每个$p\ge3$的纯$p$-自旋模型中）使速率从零点线性离开，斜率等于该原子。在临界附近，常数趋于Kondor值$9/5120$，边际区域在约束中的de Almeida-Thouless阈值处结束，并且在其首次接触之上，约束测度与Parisi测度一致至$O(\theta^5)$。所有$\beta>1$的结果使用Lopatto定理，即Parisi测度在零点累积。
英文摘要
Let $Z_N$ be the partition function of the Sherrington-Kirkpatrick model at inverse temperature $β>1$. Replica theory (Kondor near the critical temperature, Parisi and Rizzo at all temperatures) predicts that the probability that $N^{-1}\log Z_N$ exceeds its typical value by $δ$ decays like $\exp(-Naδ^{6/5})$ with an explicit $a$; Aronow and Lopatto recently proved bounds of this order. The exponent is the Legendre transform of the Parisi problem constrained to carry an atom $θ$ at zero. We show that the constrained minimum exceeds the equilibrium free energy by $c_βθ^5(1+o(1))$, where $c_β=\frac{9}{640}β^3ρ_β(0)^{-3}$ and $ρ_β(0)>0$ is the density at zero of the Parisi measure. With the fractional-moment formula this gives the exponent $\frac56(6c_β)^{-1/5}δ^{6/5}(1+o(1))$ as $N\to\infty$ and then $δ\to0$. The constrained minimizer has a marginal first positive contact, to leading order at $3θ/(2ρ_β(0))$ with an atom $θ/2$. This plateau calculus applies to every mixture with $ξ'''(0)=0$ whose Parisi measure accumulates at zero; a small field $h$ opens a gap of order $|h|^{2/3}$, and an atom of the Parisi measure at zero, present in every pure $p$-spin model with $p\ge3$, makes the rate leave zero linearly. Near criticality the constant tends to Kondor's value $9/5120$ and the marginal regime ends at a de Almeida-Thouless threshold. For the Sherrington-Kirkpatrick model at every $β>1$, a finite-step curvature inequality from an unrefereed preprint shows that every constrained minimizer either has an interval of positive contacts, all marginal, or has exactly two atoms, and that the marginal regime ends at a unique de Almeida-Thouless threshold $θ_c(β)$. The laws at every $β>1$ use that the Parisi measure accumulates at zero, a theorem of Lopatto also proved in earlier work of the author.
Comments112 pages, 4 figures, 1 table
