Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The intermediate disorder regime for stable directed polymer in Poisson environment

Min Wang

Abstract

We consider the stable directed polymer in Poisson random environment in dimension 1+1, under the intermediate disorder regime. We show that, under a diffusive scaling involving different parameters of the system, the normalized point-to-point partition function of the polymer converges in law to the solution of the stochastic heat equation with fractional Laplacian and Gaussian multiplicative noise.

The intermediate disorder regime for stable directed polymer in Poisson environment

Abstract

We consider the stable directed polymer in Poisson random environment in dimension 1+1, under the intermediate disorder regime. We show that, under a diffusive scaling involving different parameters of the system, the normalized point-to-point partition function of the polymer converges in law to the solution of the stochastic heat equation with fractional Laplacian and Gaussian multiplicative noise.
Paper Structure (9 sections, 13 theorems, 84 equations)

This paper contains 9 sections, 13 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. The model
  3. Main results
  4. Preliminaries
  5. The stochastic heat equation
  6. The Wiener--Itô chaos expansion
  7. Proofs of main theorems
  8. Proof of Theorem \ref{['main']}
  9. Proof of Theorem \ref{['main2']}

Key Result

Theorem 1.1

Under assumptions (a)(b)(c), and as $t\rightarrow \infty$, where $\omega^{v_t}$ is the Poisson point process with intensity measure $v_t dsdx$, ${\mathcal{Z}}_{\alpha, \beta^*}$ is the random variable defined by (formulaZab) in the next section.

Theorems & Definitions (17)

  • Theorem 1.1: Convergence of the normalized point-to-line partition function
  • Theorem 1.2: Convergence of the normalized point-to-point partition function
  • Proposition 2.1
  • Proposition 2.2
  • Definition 2.3: Section 4.2 in LastPenrose2017lectures
  • Definition 2.4: Definition 12.10 in LastPenrose2017lectures
  • Proposition 2.5: Corollary 12.8 in LastPenrose2017lectures
  • Proposition 2.6: Theorem 18.10 in LastPenrose2017lectures
  • Proposition 2.7
  • proof
  • ...and 7 more