Trimers, pairs and localisation on disordered random graphs
Abstract
Higher-order networks take group interactions as given. I ask what happens when a group interaction is made by pairwise dynamics on a graph, and follow it through the transition that decides whether it matters at large scale, Anderson localisation. An emergent hyperedge is a k-body bound state of a Hamiltonian whose terms live on the vertices and edges of the graph, Borromean when none of its (k−1)-body faces is bound, detected on a finite graph by a k-fold coincidence that survives the thermodynamic limit while the (k−1)-fold ones vanish. Sparse random graphs produce one from the simplest pairwise dynamics: three distinguishable particles with on-site attraction U and hopping t bind into a trimer before any pair binds. The thresholds are obtained on the Bethe lattice, in closed form for the pair, in both sectors that an expander graph has and a lattice lacks, the one a condensate sees and the one a Fermi liquid sees. On a disordered graph the emergent hyperedges nucleate on rare deep sites and localise before their parts: the compact trimer, with a narrow band and a threefold sensitivity to site energies, undergoes the Anderson transition at a disorder of 1.6t, against 17t for a single particle and 4t for a pair, and the Fermi liquid of trimers is a glass above a line between 1.5t and 1.7t, where a liquid of single particles at the same density needs 15t to 16t. The transition is the single-particle cavity transition with the composite's hopping and effective disorder. A group interaction made by pairwise dynamics is an order of magnitude more fragile to heterogeneity than the network that makes it.
Related
- A Generalized Model for Disordered Random Sequential Adsorption with Charge-Dependent Deposition
- Beyond Mutual Information: Extension Profiles and Shape Functions of Random Variable Pairs
- Scalable quantum-enhanced greedy algorithm for the maximum independent set problems on random regular graphs
- Random Cellular Automata and deBruijn Graphs
- Fuzzy Erdos-Renyi random graphs: Spectral statistics
BibTeX
@misc{vazquez2026trimers,
title = {{Trimers, pairs and localisation on disordered random graphs}},
author = {Alexei Vazquez},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32768},
eprint = {2609.32768},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32768},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Trimers, pairs and localisation on disordered random graphs
AU - Alexei Vazquez
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32768
UR - https://arxiv.org/abs/2609.32768
AB - Higher-order networks take group interactions as given. I ask what happens when a group interaction is made by pairwise dynamics on a graph, and follow it through the transition that decides whether it matters at large scale, Anderson localisation. An emergent hyperedge is a k-body bound state of a Hamiltonian whose terms live on the vertices and edges of the graph, Borromean when none of its (k−1)-body faces is bound, detected on a finite graph by a k-fold coincidence that survives the thermodynamic limit while the (k−1)-fold ones vanish. Sparse random graphs produce one from the simplest pairwise dynamics: three distinguishable particles with on-site attraction U and hopping t bind into a trimer before any pair binds. The thresholds are obtained on the Bethe lattice, in closed form for the pair, in both sectors that an expander graph has and a lattice lacks, the one a condensate sees and the one a Fermi liquid sees. On a disordered graph the emergent hyperedges nucleate on rare deep sites and localise before their parts: the compact trimer, with a narrow band and a threefold sensitivity to site energies, undergoes the Anderson transition at a disorder of 1.6t, against 17t for a single particle and 4t for a pair, and the Fermi liquid of trimers is a glass above a line between 1.5t and 1.7t, where a liquid of single particles at the same density needs 15t to 16t. The transition is the single-particle cavity transition with the composite's hopping and effective disorder. A group interaction made by pairwise dynamics is an order of magnitude more fragile to heterogeneity than the network that makes it.
ER -