论文标题
$ \ mathrm {u}（n）^2 \ times \ mathrm {o}（d）$ MULTI-MATRIX模型的多重缩放限制
Multiple scaling limits of $\mathrm{U}(N)^2 \times \mathrm{O}(D)$ multi-matrix models
论文作者
论文摘要
我们研究了复杂的多矩阵模型的双尺度和三级尺度，并使用$ \ mathrm {u}（n）^2 \ times \ times \ mathrm {o}（d）$对称。双尺度限制相当于同时服用大$ n $（矩阵大小）和大$ d $（矩阵数）限制，同时保持比率$ n/\ sqrt {d} = m $固定。三重缩放极限在于将大量$ m $限制限制，同时将耦合常数$λ$调整为其临界值$λ_c$，并保持固定的产品$ m（λ_c-λ）^α$，以某种依赖于在模型上施加的特定组合限制的$α$的$α$。我们的第一个主要结果是对任意属的Feynman图的完整递归表征，该图在双尺度限制中生存。接下来，我们将所有主要图表分类为三尺度限制，我们发现它具有带装饰的平面二进制树结构。它们的批判行为属于分支聚合物的通用类别。最后，我们将限制下三重尺度限制的所有主要图分类为三个边缘连接的（或两个粒子不可约）图。它们的批判行为属于liouville量子重力的普遍性（或者，换句话说，是布朗球体）。
We study the double- and triple-scaling limits of a complex multi-matrix model, with $\mathrm{U}(N)^2\times \mathrm{O}(D)$ symmetry. The double-scaling limit amounts to taking simultaneously the large-$N$ (matrix size) and large-$D$ (number of matrices) limits while keeping the ratio $N/\sqrt{D}=M$ fixed. The triple-scaling limit consists in taking the large-$M$ limit while tuning the coupling constant $λ$ to its critical value $λ_c$ and keeping fixed the product $M(λ_c-λ)^α$, for some value of $α$ that depends on the particular combinatorial restrictions imposed on the model. Our first main result is the complete recursive characterization of the Feynman graphs of arbitrary genus which survive in the double-scaling limit. Next, we classify all the dominant graphs in the triple-scaling limit, which we find to have a plane binary tree structure with decorations. Their critical behavior belongs to the universality class of branched polymers. Lastly, we classify all the dominant graphs in the triple-scaling limit under the restriction to three-edge connected (or two-particle irreducible) graphs. Their critical behavior falls in the universality class of Liouville quantum gravity (or, in other words, the Brownian sphere).
