On the 1/D expansion for directed polymers

TL;DR

This paper develops a variational method to analyze directed polymers in finite dimensions, revealing that corrections to mean field theory scale as D^{-4/3}, which impacts the use of 1/D expansions.

Contribution

It introduces a variational approach with a symmetrized trial function to compute finite-D corrections for directed polymers, highlighting the non-trivial D^{-4/3} scaling.

Findings

Finite-D corrections scale as D^{-4/3}

Mean field theory is exact at D=∞

The 1/D expansion requires careful application

Abstract

We present a variational approach for directed polymers in $D$ transversal dimensions which is used to compute the corrections to the mean field theory predictions with broken replica symmetry. The trial function is taken to be a symmetrized version of the mean-field solution, which is known to be exact for $D=\infty$. We compute the free energy corresponding to that function and show that the finite-$D$ corrections behave like $D^{-4/3}$. It means that the expansion in powers of 1/D should be used with great care here. We hope that the techniques developed in this note will be useful also in the study of spin glasses.