We explore in detail the properties of two melonic quantum mechanical theories which can be formulated either as fermionic matrix quantum mechanics in the new large $D$ limit, or as disordered models. Both models have a mass parameter $m$ and the transition from the perturbative large $m$ region to the strongly coupled black-hole small $m$ region is associated with several interesting phenomena. One model, with ${rm U}(n)^2$ symmetry and equivalent to complex SYK, has a line of first-order phase transitions terminating, for a strictly positive temperature, at a critical point having non-trivial, non-mean-field critical exponents for standard thermodynamical quantities. Quasi-normal frequencies, as well as Lyapunov exponents associated with out-of-time-ordered four-point functions, are also singular at the critical point, leading to interesting new critical exponents. The other model, with reduced ${rm U}(n)$ symmetry, has a quantum critical point at strictly zero temperature and positive critical mass $m_*$. For $0<m<m_*$, it flows to a new gapless IR fixed point, for which the standard scale invariance is spontaneously broken by the appearance of distinct scaling dimensions $Delta_+$ and $Delta_-$ for the Euclidean two-point function when $trightarrow +infty$ and $trightarrow -infty$ respectively. We provide several detailed and pedagogical derivations, including rigorous proofs or simplified arguments for some results that were already known in the literature.
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