Journal of Theoretical Probability

, Volume 32, Issue 1, pp 64–89 | Cite as

Nested Critical Points for a Directed Polymer on a Disordered Diamond Lattice

  • Tom AlbertsEmail author
  • Jeremy Clark


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 \(\beta \) vanishes. When \(\beta \) has the form \({\widehat{\beta }}/\sqrt{n}\) for a parameter \({\widehat{\beta }}>0\), we show that there is a cutoff value \(0< \kappa < \infty \) such that as \(n \rightarrow \infty \) the variance of the normalized partition function tends to zero for \({\widehat{\beta }}\le \kappa \) and grows without bound for \({\widehat{\beta }}> \kappa \). We obtain a more refined description of the border between these two regimes by setting the inverse temperature to \(\kappa /\sqrt{n} + \alpha _n\) where \(0 < \alpha _n \ll 1/\sqrt{n}\) and analyzing the asymptotic behavior of the variance. We show that when \(\alpha _n = \alpha (\log n-\log \log n)/n^{3/2}\) (with a small modification to deal with non-zero third moment), there is a similar cutoff value \(\eta \) for the parameter \(\alpha \) such that the variance goes to zero when \(\alpha < \eta \) and grows without bound when \(\alpha > \eta \). Extending the analysis yet again by probing around the inverse temperature \((\kappa / \sqrt{n}) + \eta (\log n-\log \log n)/n^{3/2}\), we find an infinite sequence of nested critical points for the variance behavior of the normalized partition function. In the subcritical cases \({\widehat{\beta }}\le \kappa \) and \(\alpha \le \eta \), this analysis is extended to a central limit theorem result for the fluctuations of the normalized partition function.


Statistical mechanics Directed polymers Diamond lattice Intermediate disorder Central limit theorems Nested critical points 

Mathematics Subject Classification (2010)

Primary 60F05 Secondary 60E99 82C28 



We thank an anonymous referee for several suggestions which led to a greatly improved article. Alberts gratefully acknowledges the support of Simons Foundation Collaboration Grant #351687.


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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of UtahSalt Lake CityUSA
  2. 2.Department of MathematicsUniversity of MississippiOxfordUSA

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