Finite-sized one-dimensional lazy random walks

Random walks are fundamental models for describing stochastic processes, where a particle or agent traverses along a trajectory as per the available degrees of freedom, and respecting the constraints in the system. In contrast to traditional random walks, the lazy random walks introduce a finite "resting" probability, allowing the particle to remain at its current location with a certain likelihood, rather than always hopping to adjacent sites. In this study, we explore the behavior of a lazy random walker moving across lattice sites with periodic boundary conditions, taking discrete steps in unit time intervals. We focus on the mixing time, that is, the time interval required for the probability distribution of the walker's position to spread out across all lattice sites uniformly. Our analysis employs two complementary methods: first, we transform the problem into a solvable eigenvalue framework to derive numerical insights into the mixing time; and in our second approach, we employ a probabilistic perspective that corroborates the numerical results. This dual methodology not only provides a comprehensive understanding of the relationship between mixing time and the number of lattice sites but also offers novel insights into how lazy random walks behave under periodic boundary conditions. The integration of these techniques significantly advances the current understanding of lazy random walks, distinguishing this work from prior studies on regular random walks. Our findings have broad implications for stochastic processes with delayed motion dynamics, thereby paving the way for further theoretical and applied research in this domain.