自组织临界性中的对称性涌现
Symmetry Emergence in Self-Organized Criticality
AI总结：
该研究揭示自组织临界性原型模型最大密度 regime 中仿射对称性涌现的机制，证明崩落函数标度极限为特定 Monge-Ampère 方程的唯一凹解，可精确估计密度偏差。
AI中文摘要：
我们描述了自组织临界性原型模型最大密度 regime 中仿射对称性涌现的一种机制，此时底层网格的倒数值的平方远大于根据支撑在环境凸域内部的规定概率测度分布的随机扰动点数量。此外，崩落函数（又称里程计，统计每个位点的操作次数）的适当标度极限是最优传输和微分几何领域中熟知的非线性偏微分方程的解，这使得可以精确估计任意宏观窗口内密度与其最大值的偏差。仿射对称性涌现的机制源于一个新颖的经验事实，且该事实还得到了归纳论证的支持，这些论证近期已升级为严格证明：崩落函数的标度极限是凸域上带有 Dirichlet 边界条件的 Monge-Ampère 方程的唯一凹解，其势函数为上述用作无限扰动轮廓的概率测度。
英文摘要：
We describe a mechanism of affine symmetry emergence in the maximal density regime of the prototypical model of self-organized criticality when the inverse square of the mesh of the underlying lattice is much larger than the number of random perturbation points distributed according to a prescribed probability measure supported in the interior of the ambient convex domain. Moreover, an appropriate scaling limit of the toppling function (aka odometer), which counts the number of operations per site, is a solution to a non-linear partial differential equation well known in the context of optimal transport and differential geometry, making it possible to accurately estimate the deviation of the density from its maximal value in any macroscopic window. The mechanism for the affine symmetry emergence is due to the novel empirical fact, supported in addition by inductive arguments that have recently being upgraded to a rigorous proof, that the scaling limit of the toppling function is the unique concave solution of the Monge-Ampère equation with Dirichlet boundary condition on the convex domain with the potential given by the probability measure used above as the infinite-perturbation profile.
