Crumpled glass phase of randomly polymerized membranes in the large $d$ limit

Tethered phantom membranes with quenched disorder in the internal preferred metric are studied in the limit of large embedding space dimension $d\to \infty$. We find that the instability of the flat phase previously demonstrated via $\epsilon$-expansion is towards a spin-glass-like phase which we call the crumpled glass phase. We propose a spin-glass order parameter that characterizes this phase and derive the free energy which describes the crumpled, flat and crumpled glass phases is described by local tangents which vanish on average, but display a nonzero Edwards-Anderson spin-glass order parameter. From the saddle point equations at large $d$ we obtain the equation of state, phase diagram and the exponents characterizing these phases. We estimate the effects of the higher order corrections in the $1/d$ expansion by utilizing previous results for pure membranes. We use Flory arguments to calculate the wandering exponents and discuss the relevance of self-avoidance in the crumpled glass phase