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PreprintCC BY 4.0arXiv

Quantum dynamics on the sparse Erdős-Rényi graph

  • Johannes Alt
  • Raphael Ducatez
  • Antti Knowles
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Abstract

We analyse the large-time asymptotics of the Schrödinger time evolution of a free quantum particle on the Erdős-Rényi graph 𝔾(N,d/N), where the quantum Hamiltonian is given by the adjacency matrix of 𝔾. Starting from a state localized at a vertex, the wave function is approximately radial, and we characterize precisely its asymptotic form. Our results in particular cover vertices whose degree is close to 2d, hence probing the emergence of bound states from a dynamical perspective. This provides a dynamical counterpart of the coexistence of localized and delocalized states established in [arXiv:1905.03243,arXiv:2005.14180,arXiv:2106.12519,arXiv:2109.03227,arXiv:2305.16294], with the former giving rise to a localized mode and the latter to a dispersive wave. Our results hold in the regime √(log N) ≪ d ≪ No(1), which is the full region in which such dynamical phenomena may exist, and for times 1 ≪ t ≪ TN for some explicit TN ≫ 1.

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Cite

BibTeXRIS
BibTeX
@misc{alt2026quantum,
  title = {{Quantum dynamics on the sparse Erdős-Rényi graph}},
  author = {Johannes Alt and Raphael Ducatez and Antti Knowles},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35867},
  eprint = {2610.35867},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35867},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Quantum dynamics on the sparse Erdős-Rényi graph
AU  - Johannes Alt
AU  - Raphael Ducatez
AU  - Antti Knowles
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.35867
UR  - https://arxiv.org/abs/2610.35867
AB  - We analyse the large-time asymptotics of the Schrödinger time evolution of a free quantum particle on the Erdős-Rényi graph 𝔾(N,d/N), where the quantum Hamiltonian is given by the adjacency matrix of 𝔾. Starting from a state localized at a vertex, the wave function is approximately radial, and we characterize precisely its asymptotic form. Our results in particular cover vertices whose degree is close to 2d, hence probing the emergence of bound states from a dynamical perspective. This provides a dynamical counterpart of the coexistence of localized and delocalized states established in [arXiv:1905.03243,arXiv:2005.14180,arXiv:2106.12519,arXiv:2109.03227,arXiv:2305.16294], with the former giving rise to a localized mode and the latter to a dispersive wave. Our results hold in the regime √(log N) ≪ d ≪ N^o(1), which is the full region in which such dynamical phenomena may exist, and for times 1 ≪ t ≪ T_N for some explicit T_N ≫ 1.
ER  -