Aernout van Enter, Frank den Hollander Laudatio for Michael Aizenman NAW 5/4 nr. 2 juni 2003 107 Aernout van Enter Frank den Hollander Instituut voor theoretische natuurkunde Eurandom Rijksuniversiteit Groningen Technische Universiteit Eindhoven Nijenborgh 4, 9747 AG Groningen Postbus 513, 5600 MB Eindhoven [email protected] [email protected] Laudatio Laudatio for Michael Aizenman Eens per drie jaar reikt het Wiskundig Ge- Michael is the author of seventy-five research or ‘down’) or as particles (‘occupied’ or ‘emp- nootschap in opdracht van de Koninklijke Ne- papers in journals of mathematics, physics ty’). Their finite-volume conditional distribu- derlandse Academie van Wetenschappen de and mathematical physics. He has collaborat- tions (i.e., the probabilities of events inside a Brouwermedaille uit aan een internationaal ed with many co-authors on a broad range of finite volume given the state outside) are pre- toonaangevend onderzoeker. Hij wordt uit- topics. Much of his work is inspired by proba- scribed by a nearest-neighbor interaction that genodigd om een voordracht te geven op het bility theory and statistical physics, both clas- tends to ‘align spins’ or ‘glue together parti- Nederlands Mathematisch Congres, waar na sical and quantum. In his papers he typically cles’ and that contains the temperature as a afloop de laureaat de Brouwermedaille wordt ‘rides several horses at the same time’, in the parameter. At low temperature and in two uitgereikt. In 2002 werd de medaille toege- sense that cross-fertilization between differ- or more dimensions, there exists more than kend aan Michael Aizenman voor zijn bijdra- ent areas in physics and mathematics is at one infinite-volume probability measure hav- ge aan de mathematische fysica. Op het Ne- the very heart of most of his work. ing the prescribed finite-volume conditional derlands Mathematisch Congres in Eindho- Among the many honors that Michael has probabilities. These probability measures are ven op 4 april 2002 sprak Frank den Hollan- received we mention only three: in 1990 he interpreted as the different phases of the sys- der de laudatio uit. was awarded the Norbert Wiener Award of the tem. There is a so-called ‘plus-phase’, where American Mathematical Society; since 1997 the majority of spins is up (positive magneti- The 2002 Brouwer Medal, in the field of he is a member of the National Academy of zation) or the density of particles is high (liq- mathematical physics, is awarded to Michael Sciences of the USA; since 2001 he is Editor- uid), and a ‘minus-phase’, where the majority Aizenman of Princeton University. Michael is in-Chief of Communications in Mathematical of spins is down (negative magnetization) or an outstanding mathematical physicist, who Physics, the leading international journal in the density of particles is low (gas). This phe- is admired and loved by his colleagues for his mathematical physics. nomenon is referred to as a phase transition. many beautiful contributions to the field and In his work Michael has solved a number of The temperature separating the more-phase for his warm personality. In this laudatio, we very hard open problems central to statistical region from the one-phase region is the ‘criti- give a brief overview of his career and of his mechanics and field theory. He has repeat- cal temperature’. Aizenman’s result can be in- scientific contributions. edly introduced new, powerful and elegant terpreted as the statement that below the crit- Michael studied at Hebrew University in ideas, concepts and techniques that paved ical temperature two-dimensional ‘crystals’ of Jerusalem. In 1975 he obtained his PhD at the way for others. Most of his papers are one phase inside the other phase have no Yeshiva University in New York, under the su- true eye-openers, and contain a high density facets. Various other researchers had ob- pervision of Joel Lebowitz. From 1975 to 1977 of truly new ideas. As a mathematical physi- tained partial results before him. Aizenman he was a postdoc at Princeton University, with cist pur sang he often opens up new panora- proved the definitive result. R. Dobrushin had Elliott Lieb. Both Lebowitz and Lieb are stars mas by combining ideas and techniques from earlier shown that in three or more dimen- of mathematical physics, so Michael could different parts of physics and mathematics. sions such facets do exist. hardly have done better in the choice of his teachers! From 1977 to 1982 Michael was The two-dimensional Ising model has no in- Destructive field theory in high dimensions assistant professor at Princeton University, terfaces (1980) (1981–1982) from 1982 to 1987 associate and full professor The Ising model is the paradigmatic model in Simple models of quantum field theory can be at Rutgers University in New Brunswick, and statistical physics for the study of phase tran- constructed as mathematically well-defined from 1987 to 1990 full professor at Courant In- sitions. It describes a system of two-valued objects by analytic continuation in imaginary stitute in New York. Since 1990 he is profes- random variables, living on a lattice of a cer- time of statistical-physics-like models, at or sor of Mathematics and Physics at Princeton tain dimension. These random variables can near their critical temperature and subject to University. be interpreted either as magnetic spins (‘up’ a scaling limit procedure. Aizenman proved 108 NAW 5/4 nr. 2 juni 2003 Laudatio for Michael Aizenman Aernout van Enter, Frank den Hollander that, in dimension at least four, for a class of set of eigenvalues of the random Schrodinger¨ scalar field theories, this construction cannot operator, describing the energy levels of the describe systems with interacting particles. system. Localization is modelled by the ex- Namely, the limiting quantum field theory is istence of a complete set of eigenfunctions trivial in the sense that it is Gaussian, describ- that decay exponentially fast in space. Lo- ing only non-interacting particles. Again, al- calization is expected to occur for all energies though there were partial results before, the in the case of sufficiently strong disorder, and deciding step was provided by Aizenman. for a restricted range of energies in the case of weak disorder (in three or more dimensions). Uniqueness of critical points for Ising and The work of Frohlich¨ and Spencer confirmed percolation models (with D. Barsky and R. Fer- part of this picture. Aizenman and Molchanov nández, 1985–1987) gave a different, very elegant and much sim- In percolation models, each site (or bond) plified proof of localization, based on estimat- in a lattice network is empty or occupied in ing the expectation of fractional moments of some random manner, governed by a param- resolvents of random operators with a dense eter that controls the densities of occupied point spectrum. sites (or bonds). One considers the question whether far apart sites (or bonds) are con- Conformal invariance in percolation, exis- nected by an occupied cluster and what the tence and properties of scaling limits (1995– geometry of such clusters is. Again, there is 1998) a ‘critical density’ separating a region where If one takes the scaling limit of a lattice the- clusters are finite from a region where there Photo: Pryde Brown ory in which the lattice distance approaches is an infinite cluster running through the net- zero, then one obtains a theory that lives in work. The uniqueness result showed that var- long-range Ising, Potts and percolation mod- the continuum. For many two-dimensional ious a priori different notions of critical den- els. Since this revival, a plethora of applica- statistical-physics-like models ‘at criticali- sity coincide. This work widely extended the tions has come up, including applications in ty’, the limit is predicted to be non-trivial famous result for two-dimensional indepen- numerical simulations. and to be invariant under conformal trans- dent bond percolation due to H. Kesten (1981 formations. The existence of such limits Brouwer Medal in the field of probability the- The two-dimensional random field Ising mod- and their properties are mathematically puz- ory) and solved a problem that for many years el has no phase transition (with J. Wehr, zling. Aizenman has provided various impor- was open. A similar result was proved for the 1990–1991) tant ideas, which have developed further in Ising model and various a priori different no- If one considers the Ising model in a weak the hands of other researchers, in particu- tions of critical temperature. symmetric random external magnetic field, lar, G. Lawler, O. Schramm, S. Smirnov and then the minimum dimension in which coexis- W. Werner, leading to great leaps forward in Uniqueness of infinite percolation clusters tence of different phases can occur becomes our understanding of critical phenomena in (with H. Kesten and C. Newman, 1987) three instead of two. This result was predicted two dimensions. For these developments we This result, which applies to independent in the physics literature on the basis of a non- refer to Aizenman’s Brouwer Lecture. percolation on regular lattices with a certain rigorous argument due to Y. Imry and S.-k. Ma, growth restriction, states that no two infinite but it took many years to be mathematically The above summary gives an impression (not percolation clusters can coexist, i.e., above confirmed. a full one) of the wide scope and broad in- the critical density there is one and only one terests of Aizenman’s research. His work has infinite cluster. The proof was afterwards Fractional moment estimates for the estab- been influential both among mathematicians beautifully extended and simplified by R. Bur- lishment of localization in disordered quan- and among physicists, which makes him a ton and M. Keane.