发表机构
Université Paris-Saclay; CEA; Faculté des Sciences et Techniques (ICMPA-UNESCO Chair) Université d’Abomey-Calavi（巴黎萨克雷大学; 法国原子能和替代能源委员会; 阿博米-卡拉维大学科学与技术学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出相对论背景场方法处理零维$O(N)$向量模型的RG，通过BBP相变和部分Hubbard-Stratonovich分解定义尺度，发现渐近Wilson-Fisher不动点，其临界指数与双重缩放一致。
AI 中文摘要
在最近的一篇论文（arxiv.org/abs/2609.21031）中，我们引入了相对论背景场概念，作为离散量子引力模型重整化群（RG）的一种新方法，重点研究了随机矩阵模型。所提出的方法具有双重优势：一方面保持了模型的规范对称性（通常为$U(N)$或$O(N)$），而传统方法中该对称性的破缺一直是一个未解决的问题；另一方面，它提供了对连续极限或红外（IR）极限更令人满意的处理。在该极限下，理论由维度$D = 2k + 3$中的非局域场论描述，其中$k$表示由随机矩阵理论决定的有效谱中的切割数。这篇教学性论文回顾了零维实随机向量模型的这一构造，其对称群为$O(N)$。遵循背景场型公式的一般策略，我们利用BBP（Baik-Ben Arrous-P'ech'e）相变表明，对四次相互作用进行部分Hubbard--Stratonovich分解（涉及矩阵值中间场）能够使一种优选的尺度概念出现。该尺度进而定义了RG流，我们分析了其微扰结构。我们的计算证明了存在一个渐近的Wilson--Fisher型不动点，其唯一的相关临界指数与这些模型的双重缩放临界指数在定性上一致。
英文摘要
In a recent paper arxiv.org/abs/2609.21031, we have introduced the concept of a relational background field as a new approach to the renormalization group (RG) for discrete quantum gravity models, focusing on random matrix models. The proposed method offers the dual advantage of preserving the model's gauge symmetry (typically $U(N)$ or $O(N)$), whose breaking in conventional approaches has remained an open problem, and providing a more satisfactory treatment of the continuum or infrared (IR) limit. In this limit, the theory is described by a non-local field theory in dimension $D = 2k + 3$, where $k$ denotes the number of cuts in the effective spectrum as dictated by random matrix theory. This pedagogical paper reviews this construction for real random vector models in zero dimensions, with $O(N)$ as the symmetry group. Following the general strategy of background-field-type formulations, we show, using the BBP (Baik-Ben Arrous-P'ech'e) phase transition, that a partial Hubbard--Stratonovich decomposition of the quartic interaction, involving a matrix-valued intermediate field, allows a preferred notion of scale to emerge. This scale, in turn, defines an RG flow, whose perturbative structure is analyzed. Our calculation demonstrates the existence of an \textit{asymptotic} Wilson--Fisher-type fixed point, whose sole relevant critical exponent is in qualitative agreement with the double-scaling critical exponent of these models.
Comments25 pages, 3 figures
