Universal to nonuniversal transition of the statistics of rare events during the spread of random walks

Through numerous experiments that analyzed rare event statistics in heterogeneous media, it was discovered that in many cases the probability density function for particle position, P(X,t), exhibits a slower decay rate than the Gaussian function. Typically, the decay behavior is exponential, referred to as Laplace tails. However, many systems exhibit an even slower decay rate, such as power-law, log-normal, or stretched exponential. In this study, we utilize the continuous-time random walk method to investigate the rare events in particle hopping dynamics and find that the properties of the hop size distribution induce a critical transition between the Laplace universality of rare events and a more specific, slower decay of P(X,t). Specifically, when the hop size distribution decays slower than exponential, such as e-|x|β (β>1), the Laplace universality no longer applies, and the decay is specific, influenced by a few large events, rather than by the accumulation of many smaller events that give rise to Laplace tails.