Multi-Scale Jacobi Method for Anderson Localization

A new KAM-style proof of Anderson localization is obtained. A sequence of local rotations is defined, such that off-diagonal matrix elements of the Hamiltonian are driven rapidly to zero. This leads to the first proof via multi-scale analysis of exponential decay of the eigenfunction correlator (this implies strong dynamical localization). The method has been used in recent work on many-body localization [arXiv:1403.7837].Comment: 34 pages, 8 figures, clarifications and corrections for published version; more detail in Section 4.

End-to-end Distance from the Green's Function for a Hierarchical Self-Avoiding Walk in Four Dimensions

In [BEI] we introduced a Levy process on a hierarchical lattice which is four dimensional, in the sense that the Green's function for the process equals 1/x^2. If the process is modified so as to be weakly self-repelling, it was shown that at the critical killing rate (mass-squared) \beta^c, the Green's function behaves like the free one. - Now we analyze the end-to-end distance of the model and show that its expected value grows as a constant times \sqrt{T} log^{1/8}T (1+O((log log T)/log T)), which is the same law as has been conjectured for self-avoiding walks on the simple cubic lattice Z^4. The proof uses inverse Laplace transforms to obtain the end-to-end distance from the Green's function, and requires detailed properties of the Green's function throughout a sector of the complex \beta plane. These estimates are derived in a companion paper [math-ph/0205028].Comment: 29 pages, v2: reference

New brachiopods

7 p. : ill. ; 24 cm

Dimensional Reduction for Directed Branched Polymers

Dimensional reduction occurs when the critical behavior of one system can be related to that of another system in a lower dimension. We show that this occurs for directed branched polymers (DBP) by giving an exact relationship between DBP models in D+1 dimensions and repulsive gases at negative activity in D dimensions. This implies relations between exponents of the two models: $\gamma(D+1)=\alpha(D)$ (the exponent describing the singularity of the pressure), and $\nu_{\perp}(D+1)=\nu(D)$ (the correlation length exponent of the repulsive gas). It also leads to the relation $\theta(D+1)=1+\sigma(D)$, where $\sigma(D)$ is the Yang-Lee edge exponent. We derive exact expressions for the number of DBP of size N in two dimensions.Comment: 7 pages, 1 eps figure, ref 24 correcte

Renormalization of the Higgs model: Minimizers, propagators and the stability of mean field theory

We study the effective actions S ( k ) obtained by k iterations of a renormalization transformation of the U(1) Higgs model in d =2 or 3 spacetime dimensions. We identify a quadratic approximation S Q ( k ) to S ( k ) which we call mean field theory, and which will serve as the starting point for a convergent expansion of the Green's functions, uniformly in the lattice spacing. Here we show how the approximations S Q ( k ) arise and how to handle gauge fixing, necessary for the analysis of the continuum limit. We also establish stability bounds on S Q ( k ) , uniformly in k . This is an essential step toward proving the existence of a gap in the mass spectrum and exponential decay of gauge invariant correlations.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46529/1/220_2005_Article_BF01206191.pd