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Asymptotic analysis of random walks on ice and graphite

Bernard Bercu, Fabien Montégut

Abstract

The purpose of this paper is to investigate the asymptotic behavior of random walks on three-dimensional crystal structures. We focus our attention on the 1h structure of the ice and the 2h structure of graphite. We establish the strong law of large numbers and the asymptotic normality for both random walks on ice and graphite. All our analysis relies on asymptotic results for multi-dimensional martingales.

Asymptotic analysis of random walks on ice and graphite

Abstract

The purpose of this paper is to investigate the asymptotic behavior of random walks on three-dimensional crystal structures. We focus our attention on the 1h structure of the ice and the 2h structure of graphite. We establish the strong law of large numbers and the asymptotic normality for both random walks on ice and graphite. All our analysis relies on asymptotic results for multi-dimensional martingales.

Paper Structure

This paper contains 6 sections, 4 theorems, 167 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Two possible structures
  3. Main results
  4. Proofs for the RWI
  5. Proofs for the RWG
  6. Conclusion and perspectives

Key Result

Theorem 3.1

For the RWI, we have the almost sure convergence More precisely,

Figures (7)

  • Figure 1: Hexagonal structure of the graphene
  • Figure 2: Ice with $1h$ structure.
  • Figure 3: Transition probabilities for the $1h$ structure of the ice.
  • Figure 4: Graphite with $2h$ structure.
  • Figure 5: Transition probabilities for the $2h$ structure of the graphite.
  • ...and 2 more figures

Theorems & Definitions (6)

  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Remark 3.1
  • Theorem 3.4
  • Remark 3.2