The localization transition for the directed polymer in a random environment is smooth

When d ≥ 3 , the directed polymer in a random environment on ℤ d displays a phase transition from a diffusive phase, known as weak disorder , to a localized phase, referred to as strong disorder . This transition is encoded by the behavior of the free energy of the model, defined by 𝔣 ( β ) : = lim N → ∞ ( 1 / n ) log W n β , where W n β is the normalized partition function for the directed polymer of length n . More precisely, weak disorder corresponds to 𝔣 ( β ) = 0 and strong disorder to 𝔣 ( β ) < 0 . Monotonicity and continuity of 𝔣 imply that there exists β c ∈ [ 0 , ∞ ] such that weak disorder is equivalent to β ∈ [ 0 , β c ] . Furthermore, β c > 0 if and only if d ≥ 3 . We prove that this transition is infinitely smooth in the sense that 𝔣 grows slower than any power function at the vicinity of β c , that is lim β ↓ β c log | 𝔣 ( β ) | log ( β - β c ) = ∞ .