Functional Renormalization for Random Matrix Theory: The relational background field method
Abstract
The construction of a reliable renormalization group (RG) flow for discrete gravity models that preserves their underlying symmetry group, typically U(N) or O(N), remains an open problem. For random matrix models, which are the focus of this paper, this symmetry is intrinsically tied to the interactions encoding the random geometry of two-dimensional quantum Euclidean spacetime. We develop a novel approach based on the introduction of a partial matrix-valued intermediate field. In the large-N limit, measure concentration strongly suppresses fluctuations of its singular values, allowing it to play the role of a self-consistent background field. This provides the basis for a relational RG in which the notion of scale is dynamically induced by the effective Gaussian measure in the basis where the intermediate field is diagonal. Our construction preserves the symmetry of the original model and admits a well-defined continuum limit. We show that the resulting infrared theory is described by a three-dimensional non-local Euclidean field theory with a non-trivial Wilson-Fisher-like fixed point and a single relevant direction. Remarkably, the associated critical exponent exactly matches the standard double-scaling exponent. We finally discuss extensions of the background-field approach to other discrete gravity models, such as random tensor models, and to different symmetry groups, as well as connections with more formal RG frameworks and information geometry.
Related
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- Relational background field formulation of discrete quantum gravity models: The case of O(N) vector models in zero dimensions
- Fixed points of the CKM matrix renormalization group running to all orders in perturbation theory
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BibTeX
@misc{lahoche2026functional,
title = {{Functional Renormalization for Random Matrix Theory: The relational background field method}},
author = {Vincent Lahoche and Dine Ousmane Samary},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21031},
eprint = {2609.21031},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21031},
}
RIS
TY - UNPB
TI - Functional Renormalization for Random Matrix Theory: The relational background field method
AU - Vincent Lahoche
AU - Dine Ousmane Samary
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.21031
UR - https://arxiv.org/abs/2609.21031
AB - The construction of a reliable renormalization group (RG) flow for discrete gravity models that preserves their underlying symmetry group, typically U(N) or O(N), remains an open problem. For random matrix models, which are the focus of this paper, this symmetry is intrinsically tied to the interactions encoding the random geometry of two-dimensional quantum Euclidean spacetime. We develop a novel approach based on the introduction of a partial matrix-valued intermediate field. In the large-N limit, measure concentration strongly suppresses fluctuations of its singular values, allowing it to play the role of a self-consistent background field. This provides the basis for a relational RG in which the notion of scale is dynamically induced by the effective Gaussian measure in the basis where the intermediate field is diagonal. Our construction preserves the symmetry of the original model and admits a well-defined continuum limit. We show that the resulting infrared theory is described by a three-dimensional non-local Euclidean field theory with a non-trivial Wilson-Fisher-like fixed point and a single relevant direction. Remarkably, the associated critical exponent exactly matches the standard double-scaling exponent. We finally discuss extensions of the background-field approach to other discrete gravity models, such as random tensor models, and to different symmetry groups, as well as connections with more formal RG frameworks and information geometry.
ER -