  We investigate a model of electrons with random and all-to-all hopping and
spin exchange interactions, with a constraint of no double occupancy. The model
is studied in a Sachdev-Ye-Kitaev-like large-$M$ limit with SU($M$) spin
symmetry. The saddle point equations of this model are similar to appoximate
dynamic mean field equations of realistic, non-random, $t$-$J$ models. We use
numerical studies on both real and imaginary frequency axes, along with
asymptotic analyses, to establish the existence of a critical non-Fermi-liquid
metallic ground state at large doping, with the spin correlation exponent
varying with doping. This critical solution possesses a time-reparametrization
symmetry, akin to SYK models, which contributes a linear-in-temperature
resistivity over the full range of doping where the solution is present. It is
therefore an attractive mean-field description of the overdoped region of
cuprates, where experiments have observed a linear-$T$ resistivity in a broad
region. The critical metal also displays a strong particle-hole asymmetry,
which is relevant to Seebeck coefficient measurements. We show that the
critical metal has an instability to a low-doping spin-glass phase, and compute
a critical doping value. We also describe the properties of this metallic
spin-glass phase.
