Abstract
The quantum Ising model with random couplings and random transverse fields on the Cayley tree is studied by real-space-renormalization in order to construct the whole set of eigenstates. The renormalization rules are analyzed via large deviations. The phase transition between the paramagnetic and the spin-glass many-body-localized phases involves the activated exponent and the correlation length exponent . The spin-glass-ordered cluster containing NSG spins is found to be extremely sparse with respect to the total number N of spins: its size grows only logarithmically at the critical point , and it is sub-extensive in the finite region of the spin-glass phase where the continuously varying exponent θ remains in the interval .
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