  Motivated by the pseudogap-Fermi liquid transition in doped Mott insulators,
we examine the excitations of a $t$-$J$ model with random and all-to-all
hopping and exchange. The stability of quasiparticles such as spin-1/2
fermions, spin-1 magnons, and emergent Jordan-Wigner (JW) spinless fermions is
cast as a problem of localization in the many-body Hilbert space, which is
studied by the FEAST eigensolver algorithm. At low dopings, magnons and JW
fermions are better defined than spin-1/2 fermions, which are unstable. Upon
crossing a critical value of doping around $p_c$ = 1/3, their stabilities are
interchanged. Near the critical doping, these quasiparticles are all found to
be ill-defined. The critical point is thus associated with a localization
transition in the many-body Hilbert space
