  We present numerical solutions of the spectral functions of $t$-$J$ models
with random and all-to-all exchange and global SU($M$) spin rotation symmetry.
The solutions are obtained from the saddle-point equations of the large volume
limit, followed by the large $M$ limit. These saddle point equations involve
Green's functions for fractionalized spinons and holons carrying emergent U(1)
gauge charges, obeying relations similar to those of the Sachdev-Ye-Kitaev
(SYK) models. The low frequency spectral functions are compared with an
analytic analysis of the operator scaling dimensions, with good agreement. We
also compute the low frequency and temperature behavior of gauge-invariant
observables: the electron Green's function, the local spin susceptibility and
the optical conductivity; along with the temperature dependence of the d.c.
resistivity. The time reparameterization soft mode (equivalent to the boundary
graviton in holographically dual models of two-dimensional quantum gravity)
makes important contributions to all observables, and provides a
linear-in-temperature contribution to the d.c. resistivity.
