The paper explores the application of Laplace's first law of errors to diffusive motion in various systems, including biological, glassy, and active systems. It discusses the limitations of the central limit theorem in capturing the tails of diffusive processes and introduces large deviations theory and the big jump principle to address these limitations. The authors use the continuous-time random walk (CTRW) model to analyze the behavior of particles with super-exponential and sub-exponential jump length distributions. For super-exponential distributions, they apply large deviations theory, while for sub-exponential distributions, they use the big jump principle. The paper provides a detailed analysis of these principles, including the numerical methods for sampling finite-time propagators and the Edgeworth expansion to approximate the probability density function. The authors also discuss the transition from the big jump principle to large deviations theory as the jump length distribution approaches a Gaussian form. The research highlights the importance of these theoretical frameworks in understanding the exponential decay of particle density in diffusive processes and the role of rare events in these systems.The paper explores the application of Laplace's first law of errors to diffusive motion in various systems, including biological, glassy, and active systems. It discusses the limitations of the central limit theorem in capturing the tails of diffusive processes and introduces large deviations theory and the big jump principle to address these limitations. The authors use the continuous-time random walk (CTRW) model to analyze the behavior of particles with super-exponential and sub-exponential jump length distributions. For super-exponential distributions, they apply large deviations theory, while for sub-exponential distributions, they use the big jump principle. The paper provides a detailed analysis of these principles, including the numerical methods for sampling finite-time propagators and the Edgeworth expansion to approximate the probability density function. The authors also discuss the transition from the big jump principle to large deviations theory as the jump length distribution approaches a Gaussian form. The research highlights the importance of these theoretical frameworks in understanding the exponential decay of particle density in diffusive processes and the role of rare events in these systems.