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Directed polymers in a random environment: Path localization and strong disorder. (English) Zbl 1042.60069

Directed polymers in a random environment are investigated. Using martingale decomposition and martingale analysis there are proven: equivalence between the decay rate of the partition function and some natural localization properties of the path; some quantitative estimates of the decay of the partition function in one or two dimensions, or at sufficiently low temperature; the existence of quenched free energy. In particular, the results of Carmona and Hu are generalized to general environments.

MSC:

60K37 Processes in random environments
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics