This chapter develops methods to compute asymptotics of univariate Fourier–Laplace integrals (which combine exponential decay and oscillation) and saddle point approximations. We illustrate both analytic and smooth methods for asymptotics.

This chapter derives asymptotics determined by a critical point near which the singular variety has a quadratic singularity. This necessitates introducing the theory of hyperbolic polynomials and cones of hyperbolicity, which guide advanced deformations of contours of integration on the way to computing asymptotics.

This chapter discusses assorted topics related to algebraic varieties and singular sets of multivariate rational functions. In particular, we cover Laurent expansions, polynomial amoebas, convex geometry, and bounds for generating function coefficients from so-called minimal points of singular sets.

This chapter covers standard material on generating functions in one and several variables. We describe how many common combinatorial constructions yield generating function specifications, often leading to rational or algebraic equations for generating functions. We also cover D-finite generating functions, which satisfy linear differential equations and arise both from linearly recurrent sequences and as diagonals of rational generating functions. Finally, we discuss labeled combinatorial constructions and exponential generating functions.

This chapter discusses computer algebra techniques used to apply the theorems of analytic combinatorics in several variables. We describe basic algebraic primitives, including Gröbner basis techniques, and then apply them to create algorithms certifying critical points, minimal points, Whitney stratifications, and more.

This chapter derives asymptotics determined by a critical point near which the singular variety is locally a union of smooth complex manifolds. Several explicit formulae for asymptotics are given, depending on the dimension and number of sheets meeting at the critical point.

In this chapter we investigate graph distances in preferential attachment models. We focus on typical distances as well as the diameter of preferential attachment models. We again rely on path-counting techniques, as well as local limit results. Since the local limit is a rather involved quantity, some parts of our analysis are considerably harder than those in Chapters 6 and 7.

In this chapter we investigate the distance structure of the configuration model by investigating its typical distances and its diameter. We adapt the path-counting techniques in Chapter 6 to the configuration model, and obtain typical distances from the “giant is almost local” proof. To understand the ultra-small distances for infinite-variance degree configuration models, we investigate the generation growth of infinite-mean branching processes. The relation to branching processes informally leads to the power-iteration technique that allows one to deduce typical distance results in random graphs in a relatively straightforward way.

In this chapter we investigate the connectivity structure of preferential attachment models. We start by discussing an important tool: exchangeable random variables and their distribution described in de Finetti’s Theorem. We apply these results to Pólya urn schemes, which, in turn, we use to describe the distribution of the degrees in preferential attachment models. It turns out that Pólya urn schemes can also be used to describe the local limit of preferential attachment models. A crucial ingredient is the fact that the edges in the Pólya urn representation are conditionally independent, given the appropriate randomness. The resulting local limit is the Pólya point tree, a specific multi-type branching process with continuous types.