Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Decay of the Green's function of the fractional Anderson model and connection to long-range SAW

Margherita Disertori, Roberto Maturana Escobar, Constanza Rojas-Molina

Abstract

We prove a connection between the Green's function of the fractional Anderson model and the two point function of a self-avoiding random walk with long range jumps, adapting a strategy proposed by Schenker in 2015. This connection allows us to exploit results from the theory of self-avoiding random walks to improve previous bounds known for the fractional Anderson model at strong disorder. In particular, we enlarge the range of the disorder parameter where spectral localization occurs. Moreover we prove that the decay of Green's function at strong disorder for any $0<α<1$ is arbitrarily close to the decay of the massive resolvent of the corresponding fractional Laplacian, in agreement with the case of the standard Anderson model $α=1$. We also derive upper and lower bounds for the resolvent of the discrete fractional Laplacian with arbitrary mass $m\geq 0,$ that are of independent interest.

Decay of the Green's function of the fractional Anderson model and connection to long-range SAW

Abstract

We prove a connection between the Green's function of the fractional Anderson model and the two point function of a self-avoiding random walk with long range jumps, adapting a strategy proposed by Schenker in 2015. This connection allows us to exploit results from the theory of self-avoiding random walks to improve previous bounds known for the fractional Anderson model at strong disorder. In particular, we enlarge the range of the disorder parameter where spectral localization occurs. Moreover we prove that the decay of Green's function at strong disorder for any is arbitrarily close to the decay of the massive resolvent of the corresponding fractional Laplacian, in agreement with the case of the standard Anderson model . We also derive upper and lower bounds for the resolvent of the discrete fractional Laplacian with arbitrary mass that are of independent interest.
Paper Structure (16 sections, 9 theorems, 149 equations)

This paper contains 16 sections, 9 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper.
  3. Main results and discussion
  4. Remark.
  5. Discussion of the results.
  6. Preliminary definitions and results
  7. $\tau-$regularity and apriori bound.
  8. Self-avoiding walks with long-range jumps.
  9. Comparison with a long range SAW.
  10. Properties of the fractional Laplacian.
  11. Matrix elements
  12. Resolvent decay
  13. Inverse
  14. Acknowledgements
  15. Data availability
  16. ...and 1 more sections

Key Result

Theorem 1

Assume the one site probability measure $\mathrm{P}_{0}$ is $\tau-$regular, for some $\tau\in \left(\frac{d}{d+2\upalpha },1\right)$ with $\tau-$constant $M_{\tau }(\mathrm{P}_0).$ For $s\in \left(\frac{d}{d+2\upalpha },\tau \right)$ we consider the self-avoiding walk generated by $D (x,y)=D_{\upalp Then for all $\uplambda>\lambda_{0} (s)$ and $\forall x\neq x_0\in\mathbb Z^{d}$ it holds uniforml

Theorems & Definitions (19)

  • Theorem 1
  • Theorem 2
  • proof
  • Definition 3: $\uptau-$regularity
  • Lemma 4: Decay of the SAW two point function
  • proof
  • proof : Proof of Theorem \ref{['Thm.Schenker']}
  • Proposition 5
  • proof
  • Theorem 6
  • ...and 9 more