  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 $AdS_{3}\times S^{1}$, in a generalized
scaling combining the thermodynamic and short-distance limits.
