Michael Aizenman

Localization at large disorder and at extreme energies: an elementary derivation

The work presents a short proof of localization under the conditions of either strong disorder (λ > λ0) or extreme energies for a wide class of self adjoint operators with random matrix elements, acting inl2 spaces. A prototypical example is the discrete Schrödinger operatorH=−Δ+U0(x)+λVx onZd,d≧1, withU0(x) a specified background potential and {Vx} generated as random variables. The general results apply to operators with −Δ replaced by a non-local self adjoint operatorT whose matrix elements satisfy: ∑y|Tx,y|S≦Const., uniformly inx, for somes<1. Localization means here that within a specified energy range the spectrum ofH is of the pure-point type, or equivalently — the wave functions do not spread indefinitely under the unitary time evolution generated byH. The effect is produced by strong disorder in either the potential or in the off-diagonal matrix elementsTx, y. Under rapid decay ofTx, y, the corresponding eigenfunctions are also proven to decay exponentially. The method is based on resolvent techniques. The central technical ideas include the use of low moments of the resolvent kernel, i.e. <|GE(x, y)|s> withs small enough (<1) to avoid the divergence caused by the distributions Cauchy tails, and an effective use of the simple form of the dependence ofGE(x, y) on the individual matrix elements ofH in elucidating the implications of the fundamental equation (H−E)GE(x,x0)=δx,x0. This approach simplifies previous derivations of localization results, avoiding the small denominator difficulties which have been hitherto encountered in the subject. It also yields some new results which include localization under the following sets of conditions: i) potentials with an inhomogeneous non-random partU0(x), ii) the Bethe lattice, iii) operators with very slow decay in the off-diagonal terms (Tx,y≈1/|x−y|(d+ε)), and iv) localization produced by disordered boundary conditions.

Geometric analysis of φ4 fields and Ising models. Parts I and II

We provide here the details of the proof, announced in [1], that ind>4 dimensions the (even) φ4 Euclidean field theory, with a lattice cut-off, is inevitably free in the continuum limit (in the single phase regime). The analysis is nonperturbative, and is based on a representation of the field variables (or spins in Ising systems) as source/sink creation operators in a system of random currents — which may be viewed as the mediators of correlations. In this dual representation, the onset of long-range-order is attributed to percolation in an ensemble of sourceless currents, and the physical interaction in the φ4 field — and other aspects of the critical behavior in Ising models — are directly related to the intersection properties of long current clusters. An insight into the criticality of the dimensiond=4 is derived from an analogy (foreseen by K. Symanzik) with the intersection properties of paths of Brownian motion. Other results include the proof that in certain respect, the critical behavior in Ising models is in exact agreement with the mean-field approximation in high dimensionsd>4, but not in the low dimensiond=2 — for which we establish the “universality” of hyperscaling.

Sharpness of the Phase Transition in Percolation Models

AbstractThe equality of two critical points — the percolation thresholdpH and the pointpT where the cluster size distribution ceases to decay exponentially — is proven for all translation invariant independent percolation models on homogeneousd-dimensional lattices (d≧1). The analysis is based on a pair of new nonlinear partial differential inequalities for an order parameterM(β,h), which forh=0 reduces to the percolation densityP∞ — at the bond densityp=1−e−β in the single parameter case. These are: (1)M≦h∂M/∂h+M2+βM∂M/∂β, and (2) ∂M/∂β≦|J|M∂M/∂h. Inequality (1) is intriguing in that its derivation provides yet another hint of a “ϕ3 structure” in percolation models. Moreover, through the elimination of one of its derivatives, (1) yields a pair of ordinary differential inequalities which provide information on the critical exponents

Tree graph inequalities and critical behavior in percolation models

\hat \beta

Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation

and δ. One of these resembles an Ising model inequality of Fröhlich and Sokal and yields the mean field bound δ≧2, and the other implies the result of Chayes and Chayes that

Holder Regularity and Dimension Bounds for Random Curves

\hat \beta \leqq 1