球Sherrington-Kirkpatrick模型弛豫动力学中的有限尺寸效应和自平均性损失
Finite size effects and loss of self-averageness in the relaxational dynamics of the spherical Sherrington-Kirkpatrick model
摘要：
我们重新研究了有限自由度球面Sherrington-Kirkpatrick（$p=2$）模型的梯度下降动力学.对于完全随机的初始条件，我们确定了松弛发生在三个时间区域：第一个代数区域由随机交换相互作用矩阵在其边界上的本征值分布的衰减所控制，第二个代数区域由两个极端本征值之间的间隙分布所决定，第三个代数区域由随机交换相互作用矩阵在其边界上的本征值分布的衰减所控制，第四个代数区域由随机交换相互作用矩阵在其边界上的本征值分布的衰减所控制。以及由在无序平均中采样的最小间隙确定的最终指数。我们还分析了从初始状态的放松，几乎投影在鞍的势能景观的有限大小的影响，我们表明，偏差缩放为$N^{-ν}$从完美的对齐系统逃脱的初始配置在一个时间尺度缩放为$\ln N$之后的动态不再“自平均”相对于初始条件。我们结合分析和数值方法证明了这些陈述。
英文摘要：
We revisit the gradient descent dynamics of the spherical Sherrington-Kirkpatrick ($p=2$) model with finite number of degrees of freedom. For fully random initial conditions we confirm that the relaxation takes place in three time regimes: a first algebraic one controlled by the decay of the eigenvalue distribution of the random exchange interaction matrix at its edge in the infinite size limit; a faster algebraic one determined by the distribution of the gap between the two extreme eigenvalues; and a final exponential one determined by the minimal gap sampled in the disorder average. We also analyse the finite size effects on the relaxation from initial states which are almost projected on the saddles of the potential energy landscape, and we show that for deviations scaling as $N^{-ν}$ from perfect alignment the system escapes the initial configuration in a time-scale scaling as $\ln N$ after which the dynamics no longer "self-averages" with respect to the initial conditions. We prove these statements with a combination of analytic and numerical methods.
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