The Lévy hypothesis states that inverse square Lévy walks are optimal search strategies because they maximise the encounter rate with sparse, randomly distributed, replenishable targets. It has served as a theoretical basis to interpret a wealth of experimental data at various scales, from molecular motors to animals looking for resources, putting forward the conclusion that many living organisms perform Lévy walks to explore space because of their optimal efficiency. Here we provide analytically the dependence on target density of the encounter rate of Lévy walks for any space dimension $d$ ; in particular, this scaling is shown to be {\it independent} of the Lévy exponent $α$ for the biologically relevant case $d\ge 2$, which proves that the founding result of the Lévy hypothesis is incorrect. As a consequence, we show that optimizing the encounter rate with respect to $α$ is {\it irrelevant} : it does not change the scaling with density and can lead virtually to {\it any} optimal value of $α$ depending on system dependent modeling choices. The conclusion that observed inverse square Lévy patterns are the result of a common selection process based purely on the kinetics of the search behaviour is therefore unfounded.
Lévy walkers inside spherical shells with absorbing boundaries: Towards settling the optimal Lévy walk strategy for random searches
Federal University of Rio Grande do Norte（北里奥格兰德联邦大学）；Harvard-Smithsonian Center for Astrophysics（哈佛-史密森天体物理中心）；Universidade Federal de Pernambuco（伯南布哥联邦大学）；National Institute of Science and Technology of Complex Systems, Federal University of Rio Grande do Norte（复杂系统科学与技术国家研究所，北里奥格兰德联邦大学）
Self-Avoiding Walk on the square site-diluted Ising-correlated lattice
研究了正方形格点稀释关联渗流格上的自回避行走问题。利用伊辛模型实现度量空间的空间相关性。作为一个公认的结果，（广义）弗洛里的平均场关系进行了测试，以衡量相关性的影响。在研究了微扰的Fokker-Planck方程后，我们采用了一种丰富的Rosenbluth Monte Carlo方法来研究这个问题。为了更精确地，还进行了缠绕角分析，从中提取Schramm-Loewner演化（SLE）理论（$κ$）的扩散率参数。我们发现，在临界Ising（主机）系统的指数与Flory的近似。对于非临界Ising系统，我们还发现了Walker迹的分形维数与Ising系统$ξ(T)$的关联长度的关系，即$D_F^{\text{SAW}}(T)-D_F^{\text{SAW}}(T_c)\sim \frac{1}{\sqrt{ξ(T)}}$ .
Directed walks shape a universal square-root law of entropy production rate in nonreciprocal systems
Chula Intelligent and Complex Systems Center of Excellence, Department of Physics, Faculty of Science, Chulalongkorn University（朱拉隆功大学理学院物理系智能与复杂系统卓越中心）；School of Physical and Mathematical Sciences, Nanyang Technological University（南洋理工大学物理与数学科学学院）；Institute of Theoretical Physics and Mark Kac Center for Complex Systems Research, Jagiellonian University（雅盖隆大学理论物理研究所及马克·卡克复杂系统研究中心）
Inverse square Levy walk emerging universally in goal-oriented tasks
School of Science and Engineering, Tokyo Denki University（东京电机大学理工学部）；The University of Tokyo（东京大学）；SoftBank Robotics Group Corp.（软银机器人集团公司）；Shinshu University（信州大学）；Waseda University（早稻田大学）
