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A short proof of diffusivity for the directed polymers in the weak disorder phase

Hubert Lacoin

Abstract

We provide a new short and elementary proof of diffusivity for directed polymer in a random environment in the weak disorder phase.

A short proof of diffusivity for the directed polymers in the weak disorder phase

Abstract

We provide a new short and elementary proof of diffusivity for directed polymer in a random environment in the weak disorder phase.

Paper Structure

This paper contains 11 sections, 4 theorems, 26 equations.

Table of Contents

  1. Model and results
  2. The model
  3. An invariance principle for $P^{\zeta}_n$
  4. Almost-sure subsequencial convergence
  5. A by-product of our proof
  6. Comments on main the result and its proof
  7. Proofs
  8. Main Lemma
  9. Proof of Theorem \ref{['main']}
  10. Proof of Proposition \ref{['auxil']}
  11. Proof of Proposition \ref{['clacos']}

Key Result

Proposition A

We have the following equivalence If $(i)$ and $(ii)$ hold, we further have ${\mathbb P} (W^{\zeta}_{\infty}>0)= {\mathbb P} ( \forall n>0,\ W^\zeta_n>0)$, in particular ${\mathbb P} (W^{\zeta}_{\infty}>0)=1$, if ${\mathbb P} (\zeta_{k,x}>0)=1$.

Theorems & Definitions (5)

  • Proposition A
  • Theorem 1.1
  • Proposition 1.2
  • Lemma 2.1
  • proof