• Atomic Physics
• Biophysics Theory and Experiment
• Condensed Matter Experiment
• Condensed Matter Theory
• Cosmology Experiment
• Cosmology Theory
• High Energy Experiment
• High Energy Theory
• Mathematical Physics Faculty
  • Michael Aizenman
  • Elliott H. Lieb
  • Robert Seiringer

• Particle and Nuclear Astrophysics

• Quantitative and Computational Biology
• Cosmology at Princeton
• Lewis-Sigler Institute
• Princeton Center for Theoretical Science
• Princeton Institute for the Science
  and Technology of Materials

My research has concerned the mathematical analysis of issues of physics, in particular: Disorder effects on the spectra and dynamics of operators of quantum mechanics, and related random matrix phenomena Critical behavior in systems with many degrees of freedom Scaling limits, and field theory Spin glass issues, and mathematical insights concerning competing particle systems - which are related to spin glass models through the `cavity dynamics' perspective. Typically, many of the interesting phenomena encountered in statistical mechanics are beyond the reach of exact solutions and perturbative methods. Our goal has been to develop rigorous methods which can shed light on the critical behavior even in such situations. A recurring theme has been the appearance of stochastic geometric effects which play important roles in the behavior of critical system. The work has contributed to the nascence of the current field of random fractal structures which capture the scaling limits of many critical two dimensional systems, and which bear close relations with conformal field theories. Our previous works have rigorously established the existence of model-dependent upper critical dimensions, above which the critical behavior simplifies. Works on quantum spectra and dynamics have provided new methods for dealing with localization caused by disorder, and conversely for establishing the persistence of extended states in the presence of disorder. On the latter topic the success has been rather limited; the issues continues to provide a mathematical challenge. Other goals of current work include shedding light on spectral gaps for systems with disorder, and conjectured relations with the distributions known from random matrix models. Brouwer Medal Laudatio Mathematics Department Webpage Homepage

S h o r t   L i s t   of   S e l e c t e d   P u b l i c a t i o n s:

M. Aizenman Proof of the Triviality of phi4 Field Theory and Some Mean-Field Features of Ising Models for d > 4 Phys. Rev. Lett. 47, 1 (1981) M. Aizenman Geometric Analysis of \phi^4 Fields and Ising Models Commun. Math. Phys. 86, 1 (1982) with D. Barsky Sharpness of the Phase Transition in Percolation Models Commun. Math. Phys. 108, 489 (1987) M. Aizenman Rigorous Studies of Critical Behavior Physica 140 A, 225 (1986) with J. T. Chayes, L. Chayes and C.M. Newman Discontinuity of the Order Parameter in One Dimensional 1/|x-y|^2 Ising and Potts Models J. Stat. Phys., 50 1, (1988) with J. Wehr Rounding of First-Order Phase Transitions in Systems with Quenched Disorder Phys. Rev. Lett. 62, 2503 (1989) with E.H. Lieb Magnetic Properties of Some Itinerant Electron Systems at T>0 Phys. Rev. Lett. 62, 2503 (1990) with B. Nachtergaele Geometric Aspects of Quantum Spin States Commun. Math. Phys. 164, 17 (1994) with G.M. Graf Localization Bounds for an Electron Gas J. Phys. A: Math. Gen. 31, 6783 (1998) with A. Burchard, Duke Hölder Regularity and Dimension Bounds for Random Curves Math. J. 99, 419 (1999) (A related descriptive article: http://xxx.lanl.gov/abs/cond-mat/9611040). with B. Duplantier, and A. Aharony Connectivity Exponents and External Perimeter in 2D Independent Percolation Models Phys. Rev. Lett. 83, 1359 (1999) with R. Sims, and S.L. Starr Extended variational principle for the Sherrington-Kirkpatrick spin-glass model Phys. Rev. B 68, 214403 (2003) with E.H. Lieb, R. Seiringer, J.P. Solovej, and J. Yngvason Bose-Einstein Quantum Phase Transition in an Optical Lattice Model Phys. Rev. A 70, 023612 (2004) with S. Warzel Absolutely Continuous Spectra of Quantum Tree Graphs with Weak Disorder Comm. Math. Phys. Published online in 2005 (DOI: 10.1007/s00220-005-1468-5) with R. Sims and S. Warzel Fluctuation Based Proof of the Stability of AC Spectra of Random Operators on Tree Graphs in Recent Advances in Differential Equations and Mathematical Physics, N. Chernov, Y. Karpeshina, I.W. Knowles, R.T. Lewis, and R. Weikard (eds.) AMS Contemporary Mathematics Series (to appear in 2006) (http://arxiv.org/math-ph/0510069)

O f   R e l a t e d   I n t e r e s t:

OPEN PROBLEMS IN MATHEMATICAL PHYSICS