Diffusion on a heptagonal lattice

TL;DR

This paper investigates classical and quantum diffusion on a negatively curved heptagonal lattice, revealing linear diffusion behavior due to exponential boundary growth, and compares it with diffusion on complex networks.

Contribution

It provides the first detailed analysis of diffusion processes on a heptagonal lattice, highlighting unique linear diffusion characteristics in both classical and quantum regimes.

Findings

Classical random walk displacement increases linearly with time.

Quantum diffusion on the lattice also exhibits linear behavior.

Comparison shows similarities with diffusion on complex networks.

Abstract

We study the diffusion phenomena on the negatively curved surface made up of congruent heptagons. Unlike the usual two-dimensional plane, this structure makes the boundary increase exponentially with the distance from the center, and hence the displacement of a classical random walker increases linearly in time. The diffusion of a quantum particle put on the heptagonal lattice is also studied in the framework of the tight-binding model Hamiltonian, and we again find the linear diffusion like the classical random walk. A comparison with diffusion on complex networks is also made.