Convergence of distributions on paths

TL;DR

This paper investigates how distributions on paths in weighted digraphs converge, revisiting classical results and introducing new insights using analytic combinatorics, with applications across mathematics, computer science, and system theory.

Contribution

It provides a systematic analysis of distribution convergence on paths, extending known results with novel methods from analytic combinatorics for non-strongly connected digraphs.

Findings

Revisits classical convergence results for Boltzmann and uniform distributions.

Introduces new convergence results using analytic combinatorics.

Applies findings to various fields including concurrency theory.

Abstract

We study the convergence of distributions on finite paths of weighted digraphs, namely the family of Boltzmann distributions and the sequence of uniform distributions. Targeting applications to the convergence of distributions on paths, we revisit some known results from reducible nonnegative matrix theory and obtain new ones, with a systematic use of tools from analytic combinatorics. In several fields of mathematics, computer science and system theory, including concurreny theory, one frequently faces non strongly connected weighted digraphs encoding the elements of combinatorial structures of interest; this motivates our study.