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Wednesday, April 1, 2020

**Abstract:** We will continue discussing random walks on graphs and their spectral radius, computing that the spectral radius of $\mathbb{Z}^d$ is $1$, whereas it is less than $1$ for a $d$-regular tree with $d \ge 3$. We will then consider another parameter used in studying a given random walk, namely, the expectation of the number of visits to a fixed vertex. This controls the recurrence of the random walk and we will use it to deduce that the simple random walk on $\mathbb{Z}^d$ is recurrent if and only if $d \le 2$.