The Levy walk in which the frequency of occurrence of step lengths follows a power-law distribution, can be observed in the migratory behavior of organisms at various levels. Levy walks with power exponents close to 2 are observed, and the reasons are unclear. This study aims to propose a model that universally generates inverse square Levy walks (called Cauchy walks) and to identify the conditions under which Cauchy walks appear. We demonstrate that Cauchy walks emerge universally in goal-oriented tasks. We use the term "goal-oriented" when the goal is clear, but this can be achieved in different ways, which cannot be uniquely determined. We performed a simulation in which an agent observed the data generated from a probability distribution in a two-dimensional space and successively estimated the central coordinates of that probability distribution. The agent has a model of probability distribution as a hypothesis for data-generating distribution and can modify the model such that each time a data point is observed, thereby increasing the estimated probability of occurrence of the observed data. To achieve this, the center coordinates of the model must be moved closer to those of the observed data. However, in the case of a two-dimensional space, arbitrariness arises in the direction of correction of the center; this task is goal oriented. We analyze two cases: a strategy that allocates the amount of modification randomly in the x- and y-directions, and a strategy that determines allocation such that movement is minimized. The results reveal that when a random strategy is used, the Cauchy walk appears. When the minimum strategy is used, the Brownian walk appears. The presence or absence of the constraint of minimizing the amount of movement may be a factor that causes the difference between Brownian and Levy walks.

翻译：摘要：莱维行走（步长出现频率遵循幂律分布）可在不同层级生物体的迁徙行为中观察到，其中幂指数接近2的莱维行走尤为常见，但其成因尚不明确。本研究旨在提出一个能普遍生成反平方律莱维行走（即柯西行走）的模型，并识别其出现条件。我们证明柯西行走会在目标导向任务中普遍涌现——此处"目标导向"指目标明确但实现路径多样且非唯一确定。我们进行了仿真实验：智能体在二维空间中观测服从概率分布的数据，并逐步估计该概率分布的中心坐标。智能体以概率分布模型作为数据生成分布的假设，其可在每次观测数据点时调整模型以增加观测数据的估计概率，为此需将模型中心坐标向观测数据坐标靠近。但在二维空间中，中心修正方向存在任意性，这使得任务具有目标导向性。我们分析了两种策略：随机分配x、y方向修正量的策略，以及以移动量最小化为目标的分配策略。结果表明：采用随机策略时出现柯西行走，采用最小化策略时出现布朗行走。移动量约束条件的有无可能是区分布朗行走与莱维行走的关键因素。