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We consider four-point integrals arising in the planar limit of the conformal"fishnet" theory in four dimensions. They define a two-parameter family ofhigher-loop Feynman integrals, which extend the series of ladder integrals andwere argued, based on integrability and analyticity, to admit matrix-model-likeintegral and determinantal representations. In this paper, we prove theequivalence of all these representations using exact summation and integrationtechniques. We then analyze the large-order behaviour, corresponding to thethermodynamic limit of a large fishnet graph. The saddle-point equations arefound to match known two-cut singular equations arising in matrix models,enabling us to obtain a concise parametric expression for the free-energydensity in terms of complete elliptic integrals. Interestingly, the latterdepends non-trivially on the fishnet aspect ratio and differs from a scalingformula due to Zamolodchikov for large periodic fishnets, suggesting a strongsensitivity to the boundary conditions. We also find an intriguing connectionbetween the saddle-point equation and the equation describing theFrolov-Tseytlin spinning string in $AdS_{3}\times S^{1}$, in a generalizedscaling combining the thermodynamic and short-distance limits.

Fishnet four-point integrals: integrable representations and thermodynamic limits