Random Walks: Return Probabilities in Lattice Dimensions

This lecture discusses the behavior of a particle performing a random walk on a d-dimensional square lattice. The instructor begins by introducing the concept of random walks and the probability of returning to the initial position. The lecture covers the mathematical formulation of the problem, including the probability of being at a specific site at a given time and the expected number of returns to the origin. The instructor explains the significance of dimensionality in random walks, highlighting that in one dimension, the particle is expected to return infinitely often, while in higher dimensions, the return probability decreases. The Fourier decomposition is introduced as a method to analyze the time evolution of the probability distribution. The instructor emphasizes the importance of understanding the asymptotic behavior of return probabilities as dimensions increase, leading to a comprehensive understanding of random walks in various dimensions. The lecture concludes with a discussion on the implications of these findings in the context of probability theory and statistical mechanics.