My research has concerned 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 the dynamics of competing particle systems; the topics are related through the 'cavity perspective' on SG.
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.
- 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)
- M. Aizenman and 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)
- M. Aizenman, 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)
- M. Aizenman and J. Wehr, "Rounding of First-Order Phase Transitions in Systems with Quenched Disorder", Phys. Rev. Lett. 62, 2503 (1989)
- M. Aizenman and E.H. Lieb, "Magnetic Properties of Some Itinerant Electron Systems at T>0", Phys. Rev. Lett. 62, 2503 (1990)
- M. Aizenman and B. Nachtergaele, "Geometric Aspects of Quantum Spin States", Commun. Math. Phys. 164, 17 (1994)
- M. Aizenman and G.M. Graf, "Localization Bounds for an Electron Gas", J. Phys. A: Math. Gen. 31, 6783 (1998)
- M. Aizenman and A. Burchard, "Duke Hölder Regularity and Dimension Bounds for Random Curves", Math. J. 99, 419 (1999) (A related descriptive article).
- M. Aizenman, B. Duplantier, and A. Aharony, "Connectivity Exponents and External Perimeter in 2D Independent Percolation Models", Phys. Rev. Lett. 83, 1359 (1999)
- M. Aizenman, R. Sims, and S.L. Starr, "Extended variational principle for the Sherrington-Kirkpatrick spin-glass model", Phys. Rev. B 68, 214403 (2003)
- M. Aizenman, 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)
- M. Aizenman and 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)
- M. Aizenman, 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