Papers by Michael Aizenman
arXiv (Cornell University), Dec 8, 2021
The lecture delivered at the Current Developments in Mathematics conference (Harvard-MIT, Jan. 20... more The lecture delivered at the Current Developments in Mathematics conference (Harvard-MIT, Jan. 2021) focused on the recent proof of the Gaussian structure of the scaling limits of the critical Ising and ϕ 4 fields in the marginal case of four dimensions (joint work with Hugo Duminil-Copin). These notes expand on the background of the question addressed by this result, approaching it from two partly overlapping perspectives: one concerning critical phenomena in statistical mechanics and the other functional integrals over Euclidean spaces which could serve as a springboard to quantum field theory. We start by recalling some basic results concerning the models' critical behavior in different dimensions. The analysis is framed in the models' stochastic geometric random current representation. It yields intuitive explanations as well as tools for proving a range of dimension dependent results, including: the emergence in 2D of Fermionic degrees of freedom, the nongaussianity of the scaling limits in two dimensions, and conversely the emergence of Gaussian behavior in four and higher dimensions. To cover the marginal case of 4D the tree diagram bound which has sufficed for higher dimensions needed to be supplemented by a singular correction. Its presence was established through multi-scale analysis in the recent work with HDC. Contents
arXiv (Cornell University), Apr 29, 2005
We consider radial tree extensions of one-dimensional quasi-periodic Schrödinger operators and es... more We consider radial tree extensions of one-dimensional quasi-periodic Schrödinger operators and establish the stability of their absolutely continuous spectra under weak but extensive perturbations by a random potential. The sufficiency criterion for that is the existence of Bloch-Floquet states for the one dimensional operator corresponding to the radial problem.
Graduate studies in mathematics, Dec 11, 2015
Graduate studies in mathematics, Dec 11, 2015
Graduate studies in mathematics, Dec 11, 2015
Graduate studies in mathematics, Dec 11, 2015
Graduate studies in mathematics, Dec 11, 2015
Graduate studies in mathematics, Dec 11, 2015
Graduate studies in mathematics, Dec 11, 2015
Journal of Statistical Physics, Nov 23, 2018
A streamlined derivation of the Kac-Ward formula for the planar Ising model's partition function ... more A streamlined derivation of the Kac-Ward formula for the planar Ising model's partition function is presented and applied in relating the kernel of the Kac-Ward matrices' inverse with the correlation functions of the Ising model's order-disorder correlation functions. A shortcut for both is facilitated by the Bowen-Lanford graph zeta function relation. The Kac-Ward relation is also extended here to produce a family of non planar interactions on Z 2 for which the partition function and the order-disorder correlators are solvable at special values of the coupling parameters/temperature.
Communications on Pure and Applied Mathematics, Dec 15, 2015
Discussed here are criteria for the existence of continuous components in the spectra of operator... more Discussed here are criteria for the existence of continuous components in the spectra of operators with random potential. First, the essential condition for the Simon-Wolff criterion is shown to be measurable at infinity. By implication, for the iid case and more generally potentials with the K-property the criterion is boosted by a zero-one law. The boosted criterion, combined with tunneling estimates, is then applied for sufficiency conditions for the presence of continuous spectrum for random Schrödinger operators. The general proof strategy which this yields is modeled on the resonant delocalization arguments by which continuous spectrum in the presence of disorder was previously established for random operators on tree graphs. In another application of the Simon-Wolff rank-one analysis we prove the almost sure simplicity of the pure point spectrum for operators with random potentials of conditionally continuous distribution.
For additional information and updates on this book, visit www.ams.org/bookpages/gsm-168 Library ... more For additional information and updates on this book, visit www.ams.org/bookpages/gsm-168 Library of Congress Cataloging-in-Publication Data Aizenman, Michael. Random operators : disorder effects on quantum spectra and dynamics / Michael Aizenman, Simone Warzel. pages cm.-(Graduate studies in mathematics ; volume 168) Includes bibliographical references and index.
Graduate studies in mathematics, Dec 11, 2015
Communications in Mathematical Physics, Apr 21, 2009
We consider the spectral and dynamical properties of quantum systems of n particles on the lattic... more We consider the spectral and dynamical properties of quantum systems of n particles on the lattice Z d , of arbitrary dimension, with a Hamiltonian which in addition to the kinetic term includes a random potential with iid values at the lattice sites and a finite-range interaction. Two basic parameters of the model are the strength of the disorder and the strength of the interparticle interaction. It is established here that for all n there are regimes of high disorder, and/or weak enough interactions, for which the system exhibits spectral and dynamical localization. The localization is expressed through bounds on the transition amplitudes, which are uniform in time and decay exponentially in the Hausdorff distance in the configuration space. The results are derived through the analysis of fractional moments of the n-particle Green function, and related bounds on the eigenfunction correlators.
Springer eBooks, Apr 6, 2008
I. Description of the main resu l t Col lect ive phenomena which are important features of the ph... more I. Description of the main resu l t Col lect ive phenomena which are important features of the physics of bulk systems are already exhib i ted by deceptively simple l a t t i c e models. Among such phenomena are the f i r s t order phase t rans i t i ons , which are associated with the occurance of several d i s t i n c t phases of a system in a given temperature, magnetic f i e l d and other intensive parameters. The question I would address is the p o s s i b i l i t y of a stable coexistence of phases at a phase t rans i t i on . By th is I mean the occurance of a state of an extended system which is in a thermodynamic equi l ibr ium and which in d i f f e ren t regions of the space
Moscow Mathematical Journal, 2005
We consider radial tree extensions of one-dimensional quasi-periodic Schrödinger operators and es... more We consider radial tree extensions of one-dimensional quasi-periodic Schrödinger operators and establish the stability of their absolutely continuous spectra under weak but extensive perturbations by a random potential. The sufficiency criterion for that is the existence of Bloch-Floquet states for the one dimensional operator corresponding to the radial problem.
Journal of Statistical Physics, Oct 1, 1981
The one-dimensional Coulomb system is known to have equilibrium states with nonvanishing electric... more The one-dimensional Coulomb system is known to have equilibrium states with nonvanishing electric field. These states are shown here to be analogous, and related, to the 0 vacua which have been discussed for gauge theories in two or more space-time dimensions. The system exhibits confinement of fractional charges, which we dicuss with the purpose of offering a simple example of the 0-vacua phenomenology. Precise relations and connections between onedimensional Coulomb gases and two-dimensional Abelian gauge theories, and quantum-mechanical matter systems, are discussed.
Communications in Mathematical Physics, Oct 1, 1980
Using local Ward identities we prove a number of correlation inequalities for JV-component, isotr... more Using local Ward identities we prove a number of correlation inequalities for JV-component, isotropically coupled, pair interacting ferromagnets some for all JV ^ 2 and some for JV = 2,3,4. These are used to prove a mass gap above the mean field temperature, for all JV^2. For JV = 2,3,4 we prove an upper bound on a critical exponent, and a lower bound on the susceptability which diverges as m->0.
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Papers by Michael Aizenman