2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/210561Using well-known results from statistical physics, concerning the almost-sure behavior of the free energy of directed polymers in a random medium, we prove that random tree codes achieve the distortion-rate function almost surely under a certain symmetry condition.Submitted to IEEE Trans. on Information TheoryInformation Theorytext