  We consider the continuum limit of 4d planar fishnet diagrams using
integrable spin chain methods borrowed from the $\mathcal{N}=4$
Super-Yang-Mills theory. These techniques give us control on the scaling
dimensions of single-trace operators for all values of the coupling constant in
the fishnet theory. We use them to study the thermodynamical limit of the BMN
operator corresponding to the spin chain ferromagnetic vacuum. We find that its
scaling dimension exhibits a critical behaviour when the coupling constant
approaches Zamolodchikov's critical coupling. Analysis close to that point
suggests that the continuum limit of the fishnet graphs is controlled by the
two-dimensional $AdS_{5}$ non-linear sigma model. More generally, we present
evidence that the fishnet diagrams define an integrable lattice regularization
of the $AdS_{5}$ model. A system of massless TBA equations is derived for the
tachyon energy by dualizing the TBA equations of the weakly coupled planar
$\mathcal{N}=4$ SYM theory.
