Directed polymer in γ-stable random environments

The transition from a weak-disorder (diffusive) to a strong-disorder (localized) phase for directed polymers in a random environment is a well studied phenomenon. In the most common setup, it is established that the phase transition is trivial when the transversal dimension d equals 1 or 2 (the diffusive phase is reduced to β=0) while when d≥3, there is a critical temperature βc∈(0,∞) which delimits the two phases. The proof of the existence of a diffusive regime for d≥3 is based on a second moment method (Comm. Math. Phys. 123 (1989) 529–534, Ann. Probab. 34 (2006) 1746–1770, J. Stat. Phys. 52 (1988) 609–626), and thus relies heavily on the assumption that the variable which encodes the disorder intensity (which in most of the mathematics literature assumes the form eβηx), has finite second moment. The aim of this work is to investigate how the presence/absence of phase transition may depend on the dimension d in the case when the disorder variable displays a heavier tail. To this end we replace eβηx by (1+βωx) where ωx is in the domain of attraction of a stable law with parameter γ∈(1,2). In this setup we show that a non-trivial phase transition occurs if and only if γ>1+2/d. More precisely, when γ≤1+2/d, the free energy of the system is smaller than its annealed counterpart at every temperature whereas when γ>1+2/d the martingale sequence of renormalized partition functions converges to an almost surely positive random variable for all β sufficiently small.