Random Walks: Return Probabilities in Multiple Dimensions

This lecture discusses the mathematical framework for analyzing random walks on a d-dimensional lattice. The instructor begins by defining the probability of a particle returning to its initial position after a random walk. The discussion includes the use of Fourier series to simplify the time evolution of the probability distribution. The instructor explains how to compute the return probability using the expected number of returns and introduces the concept of Fourier transforms. The lecture highlights the differences in return probabilities across dimensions, noting that in one and two dimensions, the particle will almost surely return to the origin, while in three or more dimensions, the probability of return decreases. The instructor also explores the implications of these results for understanding random walks in various physical contexts, emphasizing the importance of dimensionality in determining the behavior of the random walk. The lecture concludes with a discussion on the convergence of integrals related to these probabilities and the significance of these findings in statistical physics.