We consider four-point integrals arising in the planar limit of the conformal “fishnet” theory in four dimensions. They define a two-parameter family of higher-loop Feynman integrals, which extend the series of ladder integrals and were argued, based on integrability and analyticity, to admit matrix-model-like integral and determinantal representations. In this paper, we prove the equivalence of all these representations using exact summation and integration techniques. We then analyze the large-order behaviour, corresponding to the thermodynamic limit of a large fishnet graph. The saddle-point equations are found to match known two-cut singular equations arising in matrix models, enabling us to obtain a concise parametric expression for the free-energy density in terms of complete elliptic integrals. Interestingly, the latter depends non-trivially on the fishnet aspect ratio and differs from a scaling formula due to Zamolodchikov for large periodic fishnets, suggesting a strong sensitivity to the boundary conditions. We also find an intriguing connection between the saddle-point equation and the equation describing the Frolov-Tseytlin spinning string in AdS₃ × S¹, in a generalized scaling combining the thermodynamic and short-distance limits.

我们考虑在四维共形“渔网”理论的平面极限中产生的四点积分。他们定义了一个高环费曼积分的双参数族，它们扩展了梯形积分系列，并基于可积性和解析性进行了争论，以允许类似矩阵模型的积分和行列式表示。在本文中，我们使用精确求和和积分技术证明了所有这些表示的等价性。然后我们分析大阶行为，对应于大型渔网图的热力学极限。发现鞍点方程与矩阵模型中出现的已知二割奇异方程相匹配，使我们能够根据完全椭圆积分获得自由能密度的简明参数表达式。有趣的是，后者非平凡地取决于渔网纵横比，并且不同于由 Zamolodchikov 提出的大型周期性渔网的缩放公式，这表明对边界条件有很强的敏感性。我们还发现鞍点方程和描述 Frolov-Tseytlin 纺丝弦的方程之间存在有趣的联系。AdS ₃ × S ¹，在结合热力学和短距离限制的广义标度中。