A random walk is a probabilistic process in which the walker moves on a graph by choosing a random neighbor at every time step. Random walks eventually converge to some distribution, called the stationary distribution. Since the early 90s, the focus in the field has shifted from the question of “to what distribution does this random walk converge?” to more quantitative questions like “how fast does the random walk converge to its stationary distribution?” In studying this question, some techniques originating in the study of the heat equation has proven very useful. The talk will be focused on discussing some of the connections between these two fields through examples and proofs by pictures.