Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fluctuations in the weakly coupled 4D Anderson Hamiltonian

Simon Gabriel, Tommaso Rosati

Abstract

We study the weak coupling limit of the Anderson Hamiltonian in the critical dimension $d=4$. In a perturbative sense, we prove Gaussian fluctuations about the Green's function of the Laplacian. The fluctuations are described by an explicit effective variance, up to a critical value of the coupling constant at which we expect a phase transition in the structure of the fluctuations. The proof is based on a combinatorial analysis of Feynman diagrams, and on a detailed study of the BPHZ renormalisation of the model. We characterise the limiting distribution in terms of primitive blow-ups, and prove that no Laplacian renormalisation is present. Our approach seems applicable to a broad class of equations.

Fluctuations in the weakly coupled 4D Anderson Hamiltonian

Abstract

We study the weak coupling limit of the Anderson Hamiltonian in the critical dimension . In a perturbative sense, we prove Gaussian fluctuations about the Green's function of the Laplacian. The fluctuations are described by an explicit effective variance, up to a critical value of the coupling constant at which we expect a phase transition in the structure of the fluctuations. The proof is based on a combinatorial analysis of Feynman diagrams, and on a detailed study of the BPHZ renormalisation of the model. We characterise the limiting distribution in terms of primitive blow-ups, and prove that no Laplacian renormalisation is present. Our approach seems applicable to a broad class of equations.
Paper Structure (38 sections, 62 theorems, 441 equations)

This paper contains 38 sections, 62 theorems, 441 equations.

Table of Contents

  1. Introduction and main results
  2. Trees, Feynman diagrams, and their renormalisation
  3. Trees, Feynman diagrams and their valuation
  4. Trees as iterated stochastic integrals
  5. Feynman diagram representation of moments
  6. BPHZ renormalisation
  7. Degree, contractions, extractions, and BPHZ for Feynam diagrams
  8. The BPHZ renormalisation of trees
  9. Proof of the main results
  10. Nested bubble diagrams and their extraction sequences
  11. Proofs of Theorems \ref{['thm:main']} and \ref{['thm:main2']}
  12. Non--negative Feynman diagrams
  13. Hepp sectors and trees, and their contribution
  14. Proof of Proposition \ref{['prop_countlogs_nonneg']}
  15. Application to the Anderson model
  16. ...and 23 more sections

Key Result

Theorem 1.1

For every $n\geqslant 1$ there exists $\sigma_{n} \in (0, \infty)$ such that in distribution in $\mathcal{S}^{\prime} ( (\mathbf{T}^{4})^2 ; \mathbf{R})^{\mathbf{N}_*}$: where $( \mathrm{I}^{(n)} )_{n \geqslant 1}$ is an infinite-dimensional multivariate Gaussian, with each $\mathrm{I}^{(n)}$ equal in distribution to the field $\mathrm{I}$ described by e:Jcorr, and the coefficients $\sigma_n$ def

Theorems & Definitions (140)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Definition 2.2
  • Remark 2.3: Connection to $\Phi^{4}$--Feynman diagrams
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • ...and 130 more