2016/02/21 by Tom Alberts, Alberts, Tom, Jeremy Clark +1 · 1 citation
Mathematics · Physics and Astronomy · #60F05 #60K35 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR #msc:60F05 #msc:60K35
paper · pdf · doi:10.48550/arxiv.1602.06629
21 pages; 1 figure; We made a correction to the temperature scaling and expanded our explanations in a few of the proofs
openalex publication_date 2016/02/21 · arxiv created 2017/09/28 · arxiv updated 2017/09/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a model for a directed polymer in a random environment defined on a hierarchical diamond lattice in which i.i.d. random variables are attached to the lattice bonds. Our focus is on scaling schemes in which a size parameter n, counting the number of hierarchical layers of the system, becomes large as the inverse temperature β vanishes. When β has the form \widehatβ/√(n) for a parameter \widehatβ>0, we show that there is a cutoff value 0 < κ< ∞ such that as n → ∞ the variance of the normalized partition function tends to zero for \widehatβ≤ κ and grows without bound for \widehatβ > κ. We obtain a more refined description of the border between these two regimes by setting the inverse temperature to κ/√(n) + αn where 0 < αn ≪ 1/√(n) and analyzing the asymptotic behavior of the variance. We show that when αn = α(log n-log log n)/n3/2 (with a small modification to deal with non-zero third moment) there is a similar cutoff value η for the parameter α such that when α< η the variance goes to zero and grows without bound when α> η. Extending the analysis yet again by probing around the inverse temperature κ/√(n) + η(log n-log log n)/n3/2 we find an infinite sequence of nested critical points for the variance behavior of the normalized partition function. In the subcritical cases \widehatβ ≤ κ and α≤ η this analysis is extended to a central limit theorem result for the fluctuations of the normalized partition function.