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After reviewing the basics of the cavity method in classical systems, we showhow its quantum version, with some appropriate approximation scheme, can beused to study a system of spins with random ferromagnetic interactions and arandom transverse field. The quantum cavity equations describing theferromagnetic-paramagnetic phase transition can be transformed into thewell-known problem of a classical directed polymer in a random medium. Theglass transition of this polymer problem translates ino the existence of a`Griffith phase' close to the quantum phase transition of the quantum spinproblem, where the physics is dominated by rare events.

The cavity method for quantum disordered systems: from transverse random field ferromagnets to directed polymers in random media