论文标题
Sherrington-Kirkpatrick模型的协方差矩阵的界限
Bounds on the covariance matrix of the Sherrington-Kirkpatrick model
论文作者
论文摘要
我们考虑没有外部场和反向温度$β<1 $的Sherrington-Kirkpatrick模型，并证明Gibbs度量的协方差矩阵的预期运算符规范仅取决于$β$的常数。这回答了塔拉格兰（Talagrand）提出的一个空旷的问题，他证明了$ c（β）（\ log n）^8 $的界限。我们的结果遵循，建立了协方差矩阵的近似公式，我们通过区分TAP方程，然后最佳控制相关的错误项来获得该公式。我们通过在临界温度和低温下显示在操作员规范上的下限来补充这一结果。
We consider the Sherrington-Kirkpatrick model with no external field and inverse temperature $β<1$ and prove that the expected operator norm of the covariance matrix of the Gibbs measure is bounded by a constant depending only on $β$. This answers an open question raised by Talagrand, who proved a bound of $C(β) (\log n)^8$. Our result follows by establishing an approximate formula for the covariance matrix which we obtain by differentiating the TAP equations and then optimally controlling the associated error terms. We complement this result by showing diverging lower bounds on the operator norm, both at the critical and low temperatures.
