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The Levy walk in which the frequency of occurrence of step lengths follows apower-law distribution, can be observed in the migratory behavior of organismsat various levels. Levy walks with power exponents close to 2 are observed, andthe reasons are unclear. This study aims to propose a model that universallygenerates inverse square Levy walks (called Cauchy walks) and to identify theconditions under which Cauchy walks appear. We demonstrate that Cauchy walksemerge universally in goal-oriented tasks. We use the term "goal-oriented" whenthe goal is clear, but this can be achieved in different ways, which cannot beuniquely determined. We performed a simulation in which an agent observed thedata generated from a probability distribution in a two-dimensional space andsuccessively estimated the central coordinates of that probabilitydistribution. The agent has a model of probability distribution as a hypothesisfor data-generating distribution and can modify the model such that each time adata point is observed, thereby increasing the estimated probability ofoccurrence of the observed data. To achieve this, the center coordinates of themodel must be moved closer to those of the observed data. However, in the caseof a two-dimensional space, arbitrariness arises in the direction of correctionof the center; this task is goal oriented. We analyze two cases: a strategythat allocates the amount of modification randomly in the x- and y-directions,and a strategy that determines allocation such that movement is minimized. Theresults reveal that when a random strategy is used, the Cauchy walk appears.When the minimum strategy is used, the Brownian walk appears. The presence orabsence of the constraint of minimizing the amount of movement may be a factorthat causes the difference between Brownian and Levy walks.

Inverse square Levy walk emerging universally in goal-oriented tasks