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 beta = 0) while when d >= 3, there is a critical temperature beta(c) is an element of (0,infinity) 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 e(beta eta x)), 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 e(beta eta x) by (1 + beta omega(x)) where omega(x) is in the domain of attraction of a stable law with parameter gamma is an element of (1, 2). In this setup we show that a non-trivial phase transition occurs if and only if gamma > 1 + 2/d. More precisely, when gamma <= 1 + 2/d, the free energy of the system is smaller than its annealed counterpart at every temperature whereas when gamma > 1 + 2/d the martingale sequence of renormalized partition functions converges to an almost surely positive random variable for all beta sufficiently small.