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Coincidence of critical points for directed polymers for general environments and random walks

Stefan Junk, Hubert Lacoin

Abstract

For the directed polymer in a random environment (DPRE), two critical inverse-temperatures can be defined. The first one, $β_c$, separates the strong disorder regime (in which the normalized partition function $W^β_n$ tends to zero) from the weak disorder regime (in which $W^β_n$ converges to a nontrivial limit). The other, $\bar β_c$, delimits the very strong disorder regime (in which $W^β_n$ converges to zero exponentially fast). It was proved previously that $β_c=\bar β_c$ when the random environment is upper-bounded for the DPRE based on the simple random walk. We extend this result to general environment and arbitrary reference walk. We also prove that $β_c=0$ if and only the $L^2$-critical point is trivial.

Coincidence of critical points for directed polymers for general environments and random walks

Abstract

For the directed polymer in a random environment (DPRE), two critical inverse-temperatures can be defined. The first one, , separates the strong disorder regime (in which the normalized partition function tends to zero) from the weak disorder regime (in which converges to a nontrivial limit). The other, , delimits the very strong disorder regime (in which converges to zero exponentially fast). It was proved previously that when the random environment is upper-bounded for the DPRE based on the simple random walk. We extend this result to general environment and arbitrary reference walk. We also prove that if and only the -critical point is trivial.

Paper Structure

This paper contains 31 sections, 31 theorems, 226 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Coincidence of the critical points
  4. Integrability of $W^{\beta}_n$ at criticality
  5. Absence of phase transition in the recurrent case
  6. Organization of the paper
  7. A toolbox of preliminary results
  8. A finite volume criterion relying on fractional moments
  9. Bounding the fractional moment using the size-biased measure
  10. Spine representation for the size-biased measure
  11. Organization of the proof of Theorem \ref{['firstmain']}
  12. Identifying the right event
  13. Tail distribution of the maximum of the point-to-point partition function
  14. Proof of Proposition \ref{['spot']}
  15. Proof of Theorem \ref{['locaend']}
  16. ...and 16 more sections

Key Result

Theorem 2.1

If $\eta>0$, then $\beta_c=\bar{\beta}_c$. Furthermore, if $\beta_c>\beta_2$ then weak disorder holds at $\beta_c$.

Theorems & Definitions (52)

  • Theorem 2.1
  • Remark 2.2
  • Proposition 2.3
  • Remark 2.4
  • Corollary 2.5
  • proof
  • Proposition 2.6
  • Remark 2.7
  • Remark 2.8
  • Theorem 2.9
  • ...and 42 more