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On Partition Functions for Time-Inhomogeneous Branching Random Walks

Qianrun Wu

Abstract

We establish the phase transition and universality for the partition function of time inhomogeneous branching random walks (BRWs) with decreasing variance increment,a model related to two dimensional directed polymers. By modifying Berestycki's GMC framework (removing martingale property dependency) and adapting it to discrete BRWs, we prove that the critical value of time inhomogeneous BRWs coincides with that of time homogeneous ones, and the partition functions converge in L1 in the subcritical regime. We also extend the universality at the critical parameter, showing the same decay rate of partition functions. Our approach reveals the potential for such framework in GMC, which provides a new technical path for martingale free processes and random fields beyond log correlated. Finally, we raise some open problems related to GMC beyond log correlations, branching Brownian motions and directed polymers.

On Partition Functions for Time-Inhomogeneous Branching Random Walks

Abstract

We establish the phase transition and universality for the partition function of time inhomogeneous branching random walks (BRWs) with decreasing variance increment,a model related to two dimensional directed polymers. By modifying Berestycki's GMC framework (removing martingale property dependency) and adapting it to discrete BRWs, we prove that the critical value of time inhomogeneous BRWs coincides with that of time homogeneous ones, and the partition functions converge in L1 in the subcritical regime. We also extend the universality at the critical parameter, showing the same decay rate of partition functions. Our approach reveals the potential for such framework in GMC, which provides a new technical path for martingale free processes and random fields beyond log correlated. Finally, we raise some open problems related to GMC beyond log correlations, branching Brownian motions and directed polymers.
Paper Structure (11 sections, 15 theorems, 117 equations)

This paper contains 11 sections, 15 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Outline:
  3. Acknowledgment:
  4. The Model of Branching Random Walks and Main Results
  5. Proof of Main Results
  6. Remark:
  7. Behavior of Partition Function at Criticality $\bar{\beta}_c$
  8. Further Directions
  9. Behavior of Partition Function at Criticality $\bar{\beta}_c$
  10. Relationship with Two-Dimensional Directed Polymers
  11. Other Related Models

Key Result

Theorem 2.1

(Phase Transition for Time-homogeneous BRWs) For a BRW on a supercritical Galton–Watson tree with mean offspring number $\mathbb{E}[D_1] = d > 1$, there exists a phase transition for the non-degeneracy of $W_\infty^\beta$, i.e., is well-defined and equals $\sqrt{2\log d}$.

Theorems & Definitions (23)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • Theorem 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 13 more