Abstract
We show that for the problem of directed polymers on a tree with i.i.d. random complex weights on each bond, three possible phases can exist; the phase of a particular system is determined by the distribution ρ of the random weights. For each of these three phases, we give the expression of the free energy per unit length in the limit of infinitely long polymers. Our proofs require several hypotheses on the distribution ρ, most importantly, that the amplitude and the phase of each complex weight be statistically independent. The main steps of our proofs use bounds on noninteger moments of the partition function and self averaging properties of the free energy. We illustrate our results by some examples and discuss possible generalizations to a larger class of distributions, to Random Energy Models, and to the finite dimensional case. We note that our results are not in agreement with the predictions of a recent replica approach to a similar problem.
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References
Bolthausen, E.: A note on the diffusion of directed polymers in a random environment. Commun. Math. Phys.123, 529–534 (1989)
Buffet, E., Patrick, A., Pulé, J.V.: Directed Polymers on Trees: a Martingale Approach. J. Phys. A26, 1823–1834 (1993)
Chauvin, B., Rouault, A.: KPP equation and supercritical branching Brownian motion in the subcritical speed area. Application to spatial trees. Prob. Th. and Relat. Fields80, 299–314 (1988)
Cook, J., Derrida, B.: Polymers on disordered hierarchical lattices: A nonlinear combination of random variables. J. Stat. Phys.57, 89–139 (1989)
Cook, J., Derrida, B.: Lyapounov exponents of large, sparse random matrices and the problem of directed polymers with complex random weights. J. Stat. Phys.61, 961–986 (1990)
Dekking, F.M.: A nonlinear evolution with travelling waves. In: Luck, J.M., Moussa, P., Waldschmidt, M. (eds), Number Theory and Physics. Proceedings in Physics47, pp. 204–208. Berlin, Heidelberg, New York: Springer 1990
Derrida, B.: Random energy model: Limit of a family of disordered models. Phys. Rev. Lett.45, 79–82 (1980)
Derrida, B.: Directed polymers in a random medium. PhysicaA163, 71–84 (1990)
Derrida, B.: The zeroes of the partition function of the random energy model. PhysicaA177, 31–37 (1991)
Derrida, B., Spohn, H.: Polymers on disordered trees, spin glasses and travelling waves. J. Stat. Phys.51, 817–840 (1988)
Evans, M.R., Derrida, B.: Improved bounds for the transition temperature of directed polymers in a finite dimensional random medium. J. Stat. Phys.69, 427–437 (1992)
Goldschmidt, Y.Y., Blum, T.: Directed walks with complex random weights: Phase diagram and replica symmetry breaking. J. Phys. I (France)2, 1607–1619 (1992)
Halpin-Healy, T.: Diverse manifolds in random media. Phys. Rev. Lett.62, 442–445 (1989)
Huse, D.A., Henley, C.L.: Pinning and roughening of domain walls in Ising systems due to random impurities. Phys. Rev. Lett.54, 2708–2711 (1985)
Huse, D.A., Henley, C.L., Fisher, D.S.: Response to a comment. Phys. Rev. Lett.55, 2924–2924 (1985)
Imbrie, J.Z., Spencer, T.J.: Diffusion of directed polymers in a random environment. J. Stat. Phys.52, 609–626 (1988)
Kardar, M.: Domain walls subject to quenched impurities. J. Appl. Phys.61, 3601–3604 (1987)
Kardar, M., Parisi, G., Zhang, Y.-C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett.56, 889–892 (1986)
Kardar, M., Zhang, Y.-C.: Scaling of directed polymers in a random medium. Phys. Rev. Lett.56, 2087–2090 (1987)
Krug, J., Spohn, H.: Kinetic Roughening of Growing Surfaces. In: Godrèche, C. (ed.), Solids far from Equilibrium. Cambridge, UK: Cambridge University Press 1991
Medina, E., Kardar, M., Shapir, Y., Wang, X.R.: Interference of directed paths in disordered systems. Phys. Rev. Lett.62, 941–944 (1989)
Mezard, M., Parisi, G.: Replica field theory for random manifolds. J. Physique11, 809–836 (1991)
Moukarzel, C., Parga, N.: Numerical complex zeros of the random energy model. PhysicaA177, 24–30 (1991)
Nguyen, V.L., Spivak, B.Z., Shklovskiî, B.I.: Tunnel hopping in disordered systems. JETP Sov. Phys.62, 1021–1029 (1985)
Parisi, G.: On the replica approach to random directed polymers in two dimension. J. Physique51, 1595–1606 (1990)
Shapir, Y., Wang, X.R.: Absence ofh/e periodicity of the Aharonov-Bohm oscillations in square metallic lattices. Europhys. Lett.4, 1165–1170 (1987)
Zhang, Y.C.: Ground state instability of a random system. Phys. Rev. Lett.59, 2125–2128 (1987)
Zhang, Y.C.: Directed polymers with complex amplitudes. Phys. Rev. Lett.62, 979–979 (1989)
Zhang, Y.C.: Directed polymers with complex amplitudes. Europhys. Lett.9, 113–118 (1989)
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Communicated by T. Spencer
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Derrida, B., Evans, M.R. & Speer, E.R. Mean field theory of directed polymers with random complex weights. Commun.Math. Phys. 156, 221–244 (1993). https://doi.org/10.1007/BF02098482
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DOI: https://doi.org/10.1007/BF02098482