2025/05/19 by Hubert Lacoin, Lacoin, Hubert · 2 citations
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2505.13382
openalex publication_date 2025/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
When d≥ 3, the directed polymer a in random environment on \mathbb Zd is known to display 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 the free energy of the model, defined by \mathfrak f(β):=limN→ ∞ (1/n)log Wβn where Wβn is the normalized partition function for the directed polymer of length n. More precisely weak disorder corresponds to \mathfrak f(β)=0 and strong disorder to \mathfrak f(β)<0. Monotonicity and continuity of \mathfrak f implies that there exists βc∈ [0,∞] such that weak disorder is equivalent to β∈ [0,βc]. Furthermore βc>0 if and only if d≥ 3. We prove that this transition is infinitely smooth in the sense that \mathfrak f grows slower than any power function at the vicinity of βc, that is limβ\downarrow βc (log |\mathfrak f(β)|)/(log (β-βc))=∞.