  Models of confluent tissues are built out of tessellations of the space (both
in two and three dimensions) in which the cost function is constructed in such
a way that individual cells try to optimize their volume and surface in order
to reach a target shape. At zero temperature, many of these models exhibit a
rigidity transition that separates two phases: a liquid phase and a solid
(glassy) phase. This phenomenology is now well established but the theoretical
understanding is still not complete. In this work we consider an exactly
soluble mean field model for the rigidity transition which is based on an
abstract mapping. We replace volume and surface functions by random non-linear
functions of a large number of degrees of freedom forced to be on a compact
phase space. We then seek for a configuration of the degrees of freedom such
that these random non-linear functions all attain the same value. This target
value is a control parameter and plays the role of the target cell shape in
biological tissue models. Therefore we map the microscopic models of cells to a
random continuous constraint satisfaction problem (CCSP) with equality
constraints. We argue that at zero temperature, the rigidity transition
corresponds to the satisfiability transition of the problem. We also
characterize both the satisfiable (SAT) and unsatisfiable (UNSAT) phase. In the
SAT phase, before reaching the rigidity transition, the zero temperature SAT
landscape undergoes an RSB/ergodicity breaking transition of the same type as
the Gardner transition in amorphous solids. By solving the RSB equations we
compute the SAT/UNSAT threshold and the critical behavior around it. In the
UNSAT phase we also compute the average shape index as a function of the target
one and we compare the thermodynamical solution of the model with the results
of the numerical greedy minimization of the corresponding cost function.
