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 close 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 frequency of occurrence of the movement lengths shows a power-law distribution with exponent 2. 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：莱维行走（步长出现频率遵循幂律分布）可观察于不同层次生物的迁徙行为中。实验观测到幂指数接近2的莱维行走现象，但其成因尚不明确。本研究旨在提出一个能普适生成逆平方莱维行走（称为柯西行走）的模型，并确定柯西行走出现的条件。我们证明柯西行走在目标导向任务中会普适涌现。所谓"目标导向"，是指目标明确但达成方式存在多种可能且无法唯一确定的情形。我们进行了一项模拟实验：智能体在二维空间中观测某概率分布生成的数据点，并连续估计该概率分布的中心坐标。智能体将概率分布模型作为数据生成分布的假设，并能在每次观测数据点后修正模型，以提高对观测数据出现概率的估计。要实现这点，模型中心坐标需接近观测数据坐标。然而在二维空间中，中心修正方向存在任意性——这正是目标导向任务的特征。我们分析了两种策略：一种将修正量随机分配至x轴和y轴方向；另一种则通过最小化移动距离来确定分配方案。研究结果表明：采用随机策略时，移动步长出现频率呈幂指数为2的幂律分布；采用最小化策略时则出现布朗行走。移动量最小化约束的存在与否，可能是导致布朗行走与莱维行走差异的关键因素。