arXiv 1304.41522013-04hep-thcond-mat.stat-mechmath-phmath.MP

从矩阵模型的观点看张量模型：随机表面上的环模型的情况

Tensor models from the viewpoint of matrix models: the case of loop models on random surfaces

Valentin Bonzom, Frédéric Combes

备注：: v2: largely rewritten, new sections added, references updated

摘要：

我们研究了随机张量和随机矩阵之间的连接，通过$U(τ)$矩阵模型，产生完全包装，定向循环随机表面上。后者被发现是在双射与一组经常边着色图通常发现在张量模型。结果表明，在循环的数量的扩展是有组织的，就像秩三张量模型的1/N扩展。张量模型的最新成果进行审查和应用在这方面。例如，使环数最大化的构形正是张量模型的Melonic图，并发现了投影到Melonic扇形上的标度极限，我们还从随机曲面上的环的观点重新解释了张量模型的双标度极限.这种方法最终被推广到更高秩的张量模型，它产生的逸度$τ$的三角形尺寸$d-1$的循环。

英文摘要：

We study a connection between random tensors and random matrices through $U(τ)$ matrix models which generate fully packed, oriented loops on random surfaces. The latter are found to be in bijection with a set of regular edge-colored graphs typically found in tensor models. It is shown that the expansion in the number of loops is organized like the 1/N expansion of rank-three tensor models. Recent results on tensor models are reviewed and applied in this context. For example, configurations which maximize the number of loops are precisely the melonic graphs of tensor models and a scaling limit which projects onto the melonic sector is found. We also reinterpret the double scaling limit of tensor models from the point of view of loops on random surfaces. This approach is eventually generalized to higher-rank tensor models, which generate loops with fugacity $τ$ on triangulations in dimension $d-1$.

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英文摘要：

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