The Choquet–Deny theorem states that any random walk on a nilpotent group is Liouville. This theorem is presented and proved. We then present a recent result from 2018 by Frisch, Hartman, Tamuz, and Vahidi-Ferdowski, that these are basically the only such examples.

In this chapter we start applying the tools developed in Part I to study random walks.The notion of amenable groups is defined, and Kesten’s criterion for amenable groups is proved. We then move to define the notion of isopermitric dimension. Inequalities relating the volume growth of a group to the isoperimetric dimension and to the decay of the heat kernel are proved.

In this chapter the basic theory of Markov chains is developed, with a focus on irreducible chains.The transition matrix is introduced as well as the notions of irreducibility, periodicity, recurrence (null and positive), and transience.The theory is applied to the relationship of a random walk on a group to the random walk on a finite-index subgroup induced by the "hitting measure."

In this chapter, we look at the moments of a random variable. Specifically we demonstrate that moments capture useful information about the tail of a random variable while often being simpler to compute or at least bound. Several well-known inequalities quantify this intuition. Although they are straightforward to derive, such inequalities are surprisingly powerful. Through a range of applications, we illustrate the utility of controlling the tail of a random variable, typically by allowing one to dismiss certain “bad events” as rare. We begin by recalling the classical Markov and Chebyshev’s inequalities. Then we discuss three of the most fundamental tools in discrete probability and probabilistic combinatorics. First, we derive the complementary first and second moment methods, and give several standard applications, especially to threshold phenomena in random graphs and percolation. Then we develop the Chernoff–Cramer method, which relies on the “exponential moment” and is the building block for large deviations bounds. Two key applications in data science are briefly introduced: sparse recovery and empirical risk minimization.

In this chapter, we move on to coupling, another probabilistic technique with a wide range of applications (far beyond discrete stochastic processes). The idea behind the coupling method is deceptively simple: to compare two probability measures, it is sometimes useful to construct a joint probability space with the corresponding marginals. We begin by defining coupling formally and deriving its connection to the total variation distance through the coupling inequality. We illustrate the basic idea on a classical Poisson approximation result, which we apply to the degree sequence of an Erdos–Renyi graph. Then we introduce the concept of stochastic domination and some related correlation inequalities. We develop a key application in percolation theory. Coupling of Markov chains is the next topic, where it serves as a powerful tool to derive mixing time bounds. Finally, we end with the Chen–Stein method for Poisson approximation, a technique that applies in particular in some natural settings with dependent variables.

In this chapter, we develop spectral techniques. We highlight some applications to Markov chain mixing and network analysis. The main tools are the spectral theorem and the variational characterization of eigenvalues, which we review together with some related results. We also give a brief introduction to spectral graph theory and detail an application to community recovery. Then we apply the spectral theorem to reversible Markov chains. In particular we define the spectral gap and establish its close relationship to the mixing time. We also show in that the spectral gap can be bounded using certain isoperimetric properties of the underlying network. We prove Cheeger’s inequality, which quantifies this relationship, and introduce expander graphs, an important family of graphs with good “expansion.” Applications to mixing times are also discussed. One specific technique is the “canonical paths method,” which bounds the spectral graph by formalizing a notion of congestion in the network.