Independent Process Approximations for Random Combinatorial Structures

TL;DR

This paper investigates how well independent random processes can approximate the component structures of various random combinatorial objects, providing bounds, heuristics, and theoretical insights into their similarities and differences.

Contribution

It offers a comprehensive analysis of approximation quality, including bounds, limiting processes, and heuristics for a wide class of combinatorial structures, extending understanding of their probabilistic behavior.

Findings

Bounds for total variation distances between processes

Existence of limiting processes for approximations

Heuristics for selecting good approximations

Abstract

Many random combinatorial objects have a component structure whose joint distribution is equal to that of a process of mutually independent random variables, conditioned on the value of a weighted sum of the variables. It is interesting to compare the combinatorial structure directly to the independent discrete process, without renormalizing. The quality of approximation can often be conveniently quantified in terms of total variation distance, for functionals which observe part, but not all, of the combinatorial and independent processes. Among the examples are combinatorial assemblies (e.g. permutations, random mapping functions, and partitions of a set), multisets (e.g. polynomials over a finite field, mapping patterns and partitions of an integer), and selections (e.g. partitions of an integer into distinct parts, and square-free polynomials over finite fields). We consider issues common to all the above examples, including equalities and upper bounds for total variation distances, existence of limiting processes, heuristics for good approximations, the relation to standard generating functions, moment formulas and recursions for computing densities, refinement to the process which counts the number of parts of each possible type, the effect of further conditioning on events of moderate probability, large deviation theory and nonuniform measures on combinatorial objects, and the possibility of getting useful results by overpowering the conditioning.