Connecting finite-size scaling and renormalization-group flows in Anderson localization on random graphs
Abstract
We investigate the relation between two complementary descriptions of the Anderson transition on small-world random graphs: finite-size scaling of eigenfunction moments and renormalization-group (RG) flows of multifractal dimensions Dq. The RG flows recover the qualitative structure previously reported for the information dimension D1, while their extension to other Dq with smaller moment orders q<1/2 confirms the strongly multifractal nature of the localized phase characterized by a clear separation between multifractal and localized behaviors at strong disorder. On the other hand, we show that we can construct another beta function characterizing the flow of eigenfunction moments, whose collapse onto a single function across different disorder strengths provides a clear confirmation of single-parameter scaling property. These results indicate that the family of RG trajectories observed for the information dimension D1 need not imply two-parameter scaling and Kosterlitz-Thouless like critical behavior. To characterize these properties, one should consider different observables, such as large q>1/2 and small q<1/2 eigenfunction moments, which are associated with distinct critical behaviors, in particular different critical exponents.
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BibTeX
@misc{garciamata2026connecting,
title = {{Connecting finite-size scaling and renormalization-group flows in Anderson localization on random graphs}},
author = {Ignacio García-Mata and Weitao Chen and John Martin and Bertrand Georgeot and Gabriel Lemarié},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31872},
eprint = {2609.31872},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31872},
}
RIS
TY - UNPB
TI - Connecting finite-size scaling and renormalization-group flows in Anderson localization on random graphs
AU - Ignacio García-Mata
AU - Weitao Chen
AU - John Martin
AU - Bertrand Georgeot
AU - Gabriel Lemarié
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31872
UR - https://arxiv.org/abs/2609.31872
AB - We investigate the relation between two complementary descriptions of the Anderson transition on small-world random graphs: finite-size scaling of eigenfunction moments and renormalization-group (RG) flows of multifractal dimensions D_q. The RG flows recover the qualitative structure previously reported for the information dimension D_1, while their extension to other D_q with smaller moment orders q<1/2 confirms the strongly multifractal nature of the localized phase characterized by a clear separation between multifractal and localized behaviors at strong disorder. On the other hand, we show that we can construct another beta function characterizing the flow of eigenfunction moments, whose collapse onto a single function across different disorder strengths provides a clear confirmation of single-parameter scaling property. These results indicate that the family of RG trajectories observed for the information dimension D_1 need not imply two-parameter scaling and Kosterlitz-Thouless like critical behavior. To characterize these properties, one should consider different observables, such as large q>1/2 and small q<1/2 eigenfunction moments, which are associated with distinct critical behaviors, in particular different critical exponents.
ER -