多矩阵模型临界行为的自举研究
Bootstrapping the critical behavior of multi-matrix models
AI总结：
本文利用带正性的自举法，结合Schwinger-Dyson方程，估计了多个多矩阵模型的临界点和指数，并推测部分模型的弦磁化率指数为1/2，可能对应连续随机树极限。
摘要：
给定一个矩阵模型，通过将Schwinger-Dyson方程与其解的正性约束相结合，在大$N$极限下，人们能够获得其矩的显式和数值界限。这种技术被称为带正性的自举法。在本文中，我们使用该技术来估计几个多矩阵模型的临界点和临界指数。作为概念验证，我们首先展示它可用于发现已被充分研究的四次单矩阵模型的临界现象。然后，我们将该方法应用于几个具有各种四次相互作用的类似“未解”的2矩阵模型。我们推测并给出有力证据表明，其中一些模型的弦磁化率指数为$γ= 1/2$，这启发式地表明连续极限很可能是连续随机树。对于其他2矩阵模型，我们找到了新的弦磁化率指数的估计值，这可能表明存在新的连续极限。接着，我们研究了一个未解的3矩阵模型，它推广了具有三次相互作用的3色模型。此外，对于所有这些模型，我们能够利用Schwinger-Dyson方程的结构，在耦合常数为零时，显式地推导出大$N$极限下自由能作为幂级数展开的前几项。
英文摘要：
Given a matrix model, by combining the Schwinger-Dyson equations with positivity constraints on its solutions, in the large $N$ limit one is able to obtain explicit and numerical bounds on its moments. This technique is known as bootstrapping with positivity. In this paper we use this technique to estimate the critical points and exponents of several multi-matrix models. As a proof of concept, we first show it can be used to find the well-studied quartic single matrix model's critical phenomena. We then apply the method to several similar ``unsolved" 2-matrix models with various quartic interactions. We conjecture and present strong evidence for the string susceptibility exponent for some of these models to be $γ= 1/2$, which heuristically indicates that the continuum limit will likely be the Continuum Random Tree. For the other 2-matrix models, we find estimates of new string susceptibility exponents that may indicate a new continuum limit. We then study an unsolved 3-matrix model that generalizes the 3-colour model with cubic interactions. Additionally, for all of these models, we are able to derive explicitly the first several terms of the free energy in the large $N$ limit as a power series expansion in the coupling constants at zero by exploiting the structure of the Schwinger-Dyson equations.
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