论文标题
$ 2 $ -SPIN球形SK模型在临界温度下的自由能波动
Free energy fluctuations of the $2$-spin spherical SK model at critical temperature
论文作者
论文摘要
我们调查了$ 2 $ -Spin球形Sherrington-Kirkpatrick模型在临界温度下的自由能的波动$β_C= 1 $。当$β= 1 $时，我们发现具有方差$ \ frac {1} {1} {6n^2} \ log（n）$的渐近高斯波动，在球形案例中确认SK模型的物理学预测，并使用Ising Spins进行物理预测。我们进一步证明了在比例尺上存在关键窗口$β= 1 +α\ sqrt {\ log（n）} n^{ - 1/3} $。对于任何$α\ in \ mathbb {r} $，我们表明波动最多是$ \ sqrt {\ log（n）} / n $的，就紧密而言。如果$α\ to \ infty $以$ n \ to \ infty $的任何速率，那么，正确归一化，波动会收敛到tracy-widom $ _1 $分布。如果$α\至0 $以任何速率为$ n \ to \ infty $或$α<0 $是固定的，则波动是渐近的高斯，如$α= 0 $ case中。在确定波动时，我们将兰伯特和帕奎特的最新结果应用于光谱边缘上高斯 - $β$增强的行为。
We investigate the fluctuations of the free energy of the $2$-spin spherical Sherrington-Kirkpatrick model at critical temperature $β_c = 1$. When $β= 1$ we find asymptotic Gaussian fluctuations with variance $\frac{1}{6N^2} \log(N)$, confirming in the spherical case a physics prediction for the SK model with Ising spins. We furthermore prove the existence of a critical window on the scale $β= 1 +α\sqrt{ \log(N) } N^{-1/3}$. For any $α\in \mathbb{R}$ we show that the fluctuations are at most order $\sqrt{ \log(N) } / N$, in the sense of tightness. If $ α\to \infty$ at any rate as $N \to \infty$ then, properly normalized, the fluctuations converge to the Tracy-Widom$_1$ distribution. If $ α\to 0$ at any rate as $N \to \infty$ or $ α<0$ is fixed, the fluctuations are asymptotically Gaussian as in the $α=0$ case. In determining the fluctuations, we apply a recent result of Lambert and Paquette on the behavior of the Gaussian-$β$-ensemble at the spectral edge.
