谢林顿-柯克帕特里克模型中的淬火现象

Vittorio Erba, Freya Behrens, Florent Krzakala, Lenka Zdeborová
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摘要

谢林顿-柯克帕特里克(SK)模型是复杂的非凸能量景观的原型。一般来说,人们对在这种景观上演化并以局部达到最小值为目标的动力学过程知之甚少。在这里,我们研究了淬火,即局部旨在降低能量的动力学。我们分析了两类不同算法的收敛能量,即单旋翻转和同步动力学,重点研究了贪婪策略和勉强策略。我们对有限尺寸效应进行了精确的数值分析,并得出结论:也许与直觉相反,勉强算法与收敛到基态能量密度相容,而贪婪策略则不然。受单自旋勉强算法和贪婪算法的启发,我们研究了两种同步时间算法,即同步贪婪算法和同步勉强算法。这些同步过程可以使用动态均场理论(DMFT)和 DMFT 的新回溯版本进行分析。值得注意的是,这是首次将反向追踪 DMFT 应用于研究全连接无序模型的动态收敛特性。分析表明,同步贪心算法也能获得与基态相容的能量,而且它经历了一个动力学相变。
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Quenches in the Sherrington-Kirkpatrick model
The Sherrington-Kirkpatrick (SK) model is a prototype of a complex non-convex energy landscape. Dynamical processes evolving on such landscapes and locally aiming to reach minima are generally poorly understood. Here, we study quenches, i.e. dynamics that locally aim to decrease energy. We analyse the energy at convergence for two distinct algorithmic classes, single-spin flip and synchronous dynamics, focusing on greedy and reluctant strategies. We provide precise numerical analysis of the finite size effects and conclude that, perhaps counter-intuitively, the reluctant algorithm is compatible with converging to the ground state energy density, while the greedy strategy is not. Inspired by the single-spin reluctant and greedy algorithms, we investigate two synchronous time algorithms, the sync-greedy and sync-reluctant algorithms. These synchronous processes can be analysed using dynamical mean field theory (DMFT), and a new backtracking version of DMFT. Notably, this is the first time the backtracking DMFT is applied to study dynamical convergence properties in fully connected disordered models. The analysis suggests that the sync-greedy algorithm can also achieve energies compatible with the ground state, and that it undergoes a dynamical phase transition.
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