Limiting distributions of ergodic continuous-time quantum walks on periodic graphs
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Abstract
In this expository note, we study several families of periodic graphs that satisfy a sufficient condition for the ergodicity of the associated continuous-time quantum walk. For these graphs, we compute the limiting distribution of the walk explicitly. We uncover interesting behavior where, in some families, the walk is ergodic in both horizontal and sectional directions, while in others, ergodicity only holds in the horizontal (large N) direction. We compare this to the limiting distribution of classical random walks on the same graphs.
Abstract
In this expository note, we study several families of periodic graphs that satisfy a sufficient condition for the ergodicity of the associated continuous-time quantum walk. For these graphs, we compute the limiting distribution of the walk explicitly. We uncover interesting behavior where, in some families, the walk is ergodic in both horizontal and sectional directions, while in others, ergodicity only holds in the horizontal (large N) direction. We compare this to the limiting distribution of classical random walks on the same graphs.
Chapters in this book
- Frontmatter I
- Preface V
- Contents IX
- Soliton solutions to the Sawada–Kotera equation 1
- Changing the discrete spectrum of the half-line matrix Schrödinger operator 35
- An inverse problem on a metric graph with cycle 115
- Inverse problems for fourth-order differential operators 129
- Limiting distributions of ergodic continuous-time quantum walks on periodic graphs 141
- Some remarks on Krein–von Neumann extensions 155
- Characterization of scattering data for differential systems with a singularity 183
- On the uniform stability of an inverse problem for operators with frozen argument 223
- Non-self-adjoint Hill operators whose spectrum is a real interval 237
- On the multiple completeness of root functions of certain classes of irregular ordinary differential operator functions 257
- Inverse spectral problems for non-self-adjoint discrete operators 293
Chapters in this book
- Frontmatter I
- Preface V
- Contents IX
- Soliton solutions to the Sawada–Kotera equation 1
- Changing the discrete spectrum of the half-line matrix Schrödinger operator 35
- An inverse problem on a metric graph with cycle 115
- Inverse problems for fourth-order differential operators 129
- Limiting distributions of ergodic continuous-time quantum walks on periodic graphs 141
- Some remarks on Krein–von Neumann extensions 155
- Characterization of scattering data for differential systems with a singularity 183
- On the uniform stability of an inverse problem for operators with frozen argument 223
- Non-self-adjoint Hill operators whose spectrum is a real interval 237
- On the multiple completeness of root functions of certain classes of irregular ordinary differential operator functions 257
- Inverse spectral problems for non-self-adjoint discrete operators 293
