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Asymptotic Capacity of the Range of Random Walks on Free Products of Graphs

Lorenz A. Gilch

Abstract

In this article we prove existence of the asymptotic capacity of the range of random walks on free products of graphs. In particular, we will show that the asymptotic capacity of the range is almost surely constant and strictly positive. Furthermore, we provide a central limit theorem for the capacity of the range and show that it varies real-analytically in terms of finitely supported probability measures of constant support.

Asymptotic Capacity of the Range of Random Walks on Free Products of Graphs

Abstract

In this article we prove existence of the asymptotic capacity of the range of random walks on free products of graphs. In particular, we will show that the asymptotic capacity of the range is almost surely constant and strictly positive. Furthermore, we provide a central limit theorem for the capacity of the range and show that it varies real-analytically in terms of finitely supported probability measures of constant support.
Paper Structure (11 sections, 35 theorems, 180 equations)

This paper contains 11 sections, 35 theorems, 180 equations.

Table of Contents

  1. Introduction
  2. Random Walks on Free Products
  3. Free Products of Graphs
  4. Random Walks on Free Products
  5. Asymptotic Capacity of the Range
  6. Existence of the Asymptotic Capacity
  7. Central Limit Theorem for the Capacity of the Range
  8. Analyticity of the Asymptotic Capacity
  9. Uniform Bounds of some Generating Functions
  10. Analyticity of $\mathbb{E}[\widetilde{\mathcal{C}}_1]$
  11. Open Problems

Key Result

Theorem 1.1

Assume that $\varrho<1$. Then there exists a constant $\mathfrak{c}\in (0,1]$ such that

Theorems & Definitions (78)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Example 2.1
  • Example 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Remark 2.5
  • ...and 68 more