具有对数势的自旋玻璃中的老化现象
Aging in a spin glass with logarithmic correlations
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中文总结 AI 辅助
该研究证明，温度低于临界值时，受离散高斯自由场（DGFF）驱动的二维随机游走会在预平衡时间尺度表现出反正弦定律描述的老化现象，其源于随系统年龄增长的复杂捕获机制。
中文摘要 AI 辅助
我们研究离散二维盒上的连续时间随机游走，该游走由离散高斯自由场（DGFF）作为势驱动：当游走处于某一顶点时，其等待时间服从指数分布，均值由场的指数乘以逆温度参数给出，之后游走会以均匀随机的方式跳至相邻顶点。我们证明，当温度低于临界值时，该游走会在一系列预平衡时间尺度上表现出“老化”现象：在任意此类时间观测该游走，再经过相同量级的额外时间后，该游走仍处于距其之前位置有限距离内的概率为正，该概率渐近由广义反正弦定律给出，且该定律带有一个依赖温度的参数。我们表明，这是一种复杂捕获机制的结果，该机制会将游行走位限制在随系统年龄增长而变长的时间段内，并且我们描述了 underlying 捕获景观的复杂结构，其与 DGFF 的近极值水平集几何结构密切相关。总体而言，本研究首次在具有对数相关势的自旋玻璃型系统的整个玻璃态阶段中，证明了如物理文献所预测的反正弦定律老化现象。
英文摘要
We consider a continuous-time random walk on the discrete two dimensional box, driven by the discrete Gaussian Free Field (DGFF) acting as potential: When at a vertex, the walk waits an exponentially distributed time with mean given by the exponential of the field times an inverse temperature parameter and then jumps to one of its neighbors uniformly at random. We prove that when the temperature is below the critical value the walk exhibits ``aging'' at a range of pre-equilibrium time scales: Observed at any such time and then again after an additional time of the same order, there is a positive probability that the walk is found within finite distance from where it was before, with this probability given asymptotically by the Generalized Arcsine Law with a temperature-dependent parameter. We show that this is a consequence of an intricate trapping mechanism which localizes the walk for periods of time which increase with the age of the system, and describe the complex structure of the underlying trapping landscape, which is intimately related to the geometry of the near-extreme level-sets of the DGFF. Altogether, this work demonstrates for the first time an Arcsine-Law aging in the case of a spin-glass-type system with a logarithmically correlated potential, throughout its glassy phase, as predicted in the physic literature.
