Center for Applied Mathematical Sciences Distinguished Lecturer, Spring 2017

Center for Applied Mathematical Sciences Distinguished Lecturer, Spring 2017 Michael Aizenman Princeton University Stochastic Geometry of Correlations in Stat-Mech and Quantum Systems Abstract: Some of the qualitative features of interacting classical and quantum systems can be illuminated through stochastic geometric representations. In these, the correlations in some of the basic model are presented as mediated through fluctuating clusters and/or random loops. Such representations facilitate insights on a number of phenomena, including: existence of phase transitions related to the onset of long range order, dimension dependence of the critical exponents in Ising type models, the emergence of conformal invariance in critical two dimensional models and relations with the conformally invariant SLE random curves. For one dimensional quantum spin chains a stochastic geometric representation allows us to shed light on the difference between the integer and half integer cases in the spectral (Haldane) gap. In the talk we tread on grounds which were earlier marked by USC Professor Mark Kac, to whose memory this lecture is dedicated. Monday, April 3, 2017 Michael Aizenman is a Professor of Physics and Kaprielian Hall Mathematics at Princeton University. His research concerns topics which present challenges and Reception: 3:00 p.m., KAP 410 opportunities for analysis and probability theory, which are however motivated by their relevance Lecture: 3:30 p.m., KAP 414 for physics. His many honors and awards include the AMS-SIAM Norbert Wiener Award (1990), the Wine & Cheese: 4:30 p.m., KAP 410 Brower Medal (2002), Doctor Honoris Causa from the University de Cergy Pontoise (2009), the APS Heineman Prize (2010), the Simons Fellowship in Mathematics (2012). He is a Fellow of the AMS and a Member of the National Academy of Sciences. CAMS Director: Susan Friedlander [email protected] .

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Spring 2019 Fine Letters
Spring 2019 • Issue 8 Department of MATHEMATICS Princeton University From the Chair Professor Allan Sly Receives MacArthur Fellowship Congratulations to Sly works on an area of probability retical computer science, where a key the Class of 2019 theory with applications from the goal often is to understand whether and all the finishing physics of magnetic materials to it is likely or unlikely that a large set graduate students. computer science and information of randomly imposed constraints on a Congratulations theory. His work investigates thresh- system can be satisfied. Sly has shown to the members of olds at which complex networks mathematically how such systems of- class of 2018 and suddenly change from having one ten reach a threshold at which solving new Ph. D.s who set of properties to another. Such a particular problem shifts from likely are reading Fine Letters for the first questions originally arose in phys- or unlikely. Sly has used a party invi- time as alumni. As we all know, the ics, where scientists observed such tation list as an analogy for the work: Math major is a great foundation for shifts in the magnetism of certain As you add interpersonal conflicts a diverse range of endeavors. This metal alloys. Sly provided a rigorous among a group of potential guests, it is exemplified by seventeen '18's who mathematical proof of the shift and can suddenly become effectively im- have gone to industry and seventeen a framework for identifying when possible to create a workable party. to grad school; ten to advanced study such shifts occur.
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Laudatio for Michael Aizenman NAW 5/4 Nr
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.
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A Life in Statistical Mechanics Part 1: from Chedar in Taceva to Yeshiva University in New York
Eur. Phys. J. H 42, 1–21 (2017) DOI: 10.1140/epjh/e2017-80006-9 THE EUROPEAN PHYSICAL JOURNAL H Oral history interview A life in statistical mechanics Part 1: From Chedar in Taceva to Yeshiva University in New York Joel L. Lebowitz1,a and Luisa Bonolis2,b 1 Departments of Mathematics and Physics, Rutgers, The State University, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA 2 Max Planck Institute for the History of Science, Boltzmannstrasse 22, 14195 Berlin, Germany Received 10 February 2017 / Accepted 10 February 2017 Published online 4 April 2017 c The Author(s) 2017. This article is published with open access at Springerlink.com Abstract. This is the ﬁrst part of an oral history interview on the life- long involvement of Joel Lebowitz in the development of statistical mechanics. Here the covered topics include the formative years, which overlapped the tragic period of Nazi power and World War II in Eu- rope, the emigration to the United States in 1946 and the schooling there. It also includes the beginnings and early scientiﬁc works with Peter Bergmann, Oliver Penrose and many others. The second part will appear in a forthcoming issue of Eur. Phys. J. H. 1 From war ravaged Europe to New York L. B. Let’s start from the very beginning. Where were you born? J. L. I was born in Taceva, a small town in the Carpathian mountains, in an area which was at that time part of Czechoslovakia, on the border of Romania and about a hundred kilometers from Poland. That was in 1930, and the town then had a population of about ten thousand people, a small town, but fairly advanced.
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09W5055 Statistical Mechanics on Random Structures
09w5055 Statistical Mechanics on Random Structures Anton Bovier (Rheinische Friedrich-Wilhelms-Universitat¨ Bonn), Pierluigi Contucci (University of Bologna), Frank den Hollander (University of Leiden and EURANDOM), Cristian Giardina` (TU Eindhoven and EURANDOM). 15 November - 20 November 2009 1 Overview of the Field The theme of the workshop has been equilibrium and non-equilibrium statistical mechanics in a random spatial setting. Put differently, the question was what happens when the world of interacting particle systems is put together with the world of disordered media. This area of research is lively and thriving, with a constant flow of new ideas and exciting developments, in the best of the tradition of mathematical physics. Spin glasses were at the core of the program, but in a broad sense. Spin glass theory has found ap- plications in a wide range of areas, including information theory, coding theory, algorithmics, complexity, random networks, population genetics, epidemiology and finance. This opens up many new challenges to mathematics. 2 Recent Developments and Open Problems The workshop brought together researchers whose interest lies at the intersection of disordered statistical mechanics and random graph theory, with a clear emphasis on applications. The multidisciplinary nature of the proposed topics has attracted research groups with different backgrounds and thus provided exchange of ideas with cross-fertilisation. As an example, we mention two problems on which we focused during the workshop. The first problem has its origin in the many fundamental issues that are still open in the theory of spin glasses. Even tough today we have a rigorous proof, in the context of mean-field models, of the solution for the free energy first proposed by G.
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A Mathematical Physicist's Perspective on Statistical Mechanics
A mathematical physicist’s perspective on Statistical Mechanics Michael Aizenman Princeton University André Aisenstadt Lecture (I) CRM, Montreal Sept. 24, 2018 1 /14 Statistical mechanics explains and quantiﬁes the process by which structure emerges from chaos. Its genesis is in Boltzmann’s explanation of thermodynamical behavior and in particular of the concept of entropy. The statistic mechanical perspective was instrumental for Planck’s theory of the light quantization and Einstein’s calculation of the Avogadro number. More recent developments include links between the physics of critical phenomena and the mathematics of conformally invariant random structures, stochastic integrability, and representation theory. The talk will focus on examples of observations and conjectures which turned out to point in fruitful directions. 2 /14 Statistical Mechanics: laws emerging from chaos Laws expressed in equations F = ma, E = mc2, PV = nRT . A bird of a seemingly different feather: ∆S ≥ 0 . Q: what is entropy? L. Boltzmann: Thermodynamics emerges from Statistical Mechanics! Stat-Mech starts with a quantiﬁcation of chaos: S = kB log W StatMech perspective was embraced and used for further developments by: M. Planck ) quantum theory of light (surmised from the black body radiation) A. Einstein ) Avogadro number from Brownian motion (exp. Perrin) −itH R. Feynman ) path representation of quantum dynamics: Ψt = e Ψ0 J. Wheeler: “There is no law except the law that there is no law.” Paraphrased: all physics laws are emergent features. 3 /14 Statistical Mechanics: laws emerging from chaos Laws expressed in equations F = ma, E = mc2, PV = nRT . A bird of a seemingly different feather: ∆S ≥ 0 . Q: what is entropy? L.
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On Spin Systems with Quenched Randomness: Classical and Quantum
On Spin Systems with Quenched Randomness: Classical and Quantum Rafael L. Greenblatt(a) ∗ Michael Aizenman(b) y Joel L. Lebowitz(c)∗ (a) Department of Physics and Astronomy Rutgers University, Piscataway NJ 08854-8019, USA (b) Departments of Physics and Mathematics Princeton University, Princeton NJ 08544, USA (a) Departments of Mathematics and Physics Rutgers University, Piscataway NJ 08854-8019, USA December 5, 2009 Dedicated to Nihat Berker on the occasion of his 60th birthday Abstract The rounding of ﬁrst order phase transitions by quenched randomness is stated in a form which is applicable to both classical and quantum systems: The free energy, as well as the ground state energy, of a spin system on a d-dimensional lattice is continuously differentiable with respect to any parameter in the Hamiltonian to which some randomness has been added when d ≤ 2. This implies absence of jumps in the associated order parameter, e.g., the magnetization in case of a random magnetic ﬁeld. A similar result applies in cases of continuous symmetry breaking for d ≤ 4. Some questions concerning the behavior of related order parameters in such random systems are discussed. 1 Introduction The effect of quenched randomness on the equilibrium and transport properties of macroscopic systems is a subject of great theoretical and practical interest which is close to Nihat’s heart. He and his collaborators [1, 2, 3] made important contributions to the study of the changes brought about by such randomness in phase transitions occurring in the pure (non-random) system. These effects can be profound in low dimensions. Their understanding evolved in a somewhat interesting way.
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Curriculum Vitae
Jack Hanson City College of NY [email protected] Dept. of Mathematics 1 (212) 650 - 5174 NAC 6/292 Appointments • Assistant Professor, City College of NY Dept. of Mathematics, 01/2016{present • Visiting Assistant Professor, GA Tech Dept. of Mathematics, 8/2015{01/2016 • Zorn Postdoctoral Fellow, Indiana University Dept. of Mathematics. 8/2013{7/2015 Education • Ph.D., Physics, Princeton University, June 2013. Advisor: Michael Aizenman • M.A., Physics, Princeton University, 2010. • B.S. Summa cum Laude, Physics, Rutgers University, 2008. Double major, Physics and Mathematics. Funding • NSF Grant DMS-1612921, \Correlations and Scaling in Disordered and Critical Stochastic Models", 2016 { 2019. $100,000. • (Funded but declined in favor of NSF) NSA Young Investigator Grant, 2016 { 2018. $40,000. • AMS-Simons travel grant, 2015 { 2016. • NSF Graduate Research Fellow, 2008 - 2013. Talks • Upcoming, untitled { Conference on Random Walks, Random Graphs and Random Media at LMU, Munich, 9 / 2019 • \Universality of the time constant for critical first-passage percolation on the triangular lat- tice" { MIT Probability Seminar, 3/2019 1 { CUNY Probability Seminar, 3/2019 { Ohio State U. Probability Seminar, 4/2019 • \Geodesics in First-Passage Percolation" { Montréalsummer workshop on challenges in probability and mathematical physics, 7/2018. • \Half-space critical exponents in high-dimensional percolation." { Conference on Recent Trends in Continuous and Discrete Probability, GA Tech, 6/2018. { U. of Rochester Probability Seminar, 9/2018 { Columbia U. Probability Seminar, 2/2019 { NYU Probability Seminar, 2/2019 • \The boundary in first-passage percolation" { Cornell Probability Seminar, 11/2017 • \Strict inequality for the chemical distance exponent in 2d percolation." { Stockholm University Probability/Statistics seminar, 8/2017 { CUNY Probability seminar, 9/2017 • \Geodesics and Busemann functions in first-passage percolation" { AMS short course, Atlanta, 1/2017.
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Arxiv:1104.3515V3
Proof of Rounding by Quenched Disorder of First Order Transitions in Low-Dimensional Quantum Systems Michael Aizenman∗ Departments of Physics and Mathematics, Princeton University, Princeton NJ 08544-8019 Rafael L. Greenblatt† Dipartamento di Matematica, Universitàdegli Studi Roma Tre, Largo San Leonardo Murialdo 1, 00146 Roma, Italy Joel L. Lebowitz‡ Departments of Mathematics and Physics, Rutgers University, Piscataway NJ 08854-8019 (Dated: November 23, 2011) Abstract We prove that for quantum lattice systems in d ≤ 2 dimensions the addition of quenched disorder rounds any first order phase transition in the corresponding conjugate order parameter, both at positive temperatures and at T = 0. For systems with continuous symmetry the statement extends up to d ≤ 4 dimensions. This establishes for quantum systems the existence of the Imry-Ma phenomenon which for classical systems was proven by Aizenman and Wehr. The extension of the proof to quantum systems is achieved by carrying out the analysis at the level of thermodynamic quantities rather than equilibrium states. PACS numbers: 75.10.Jm,64.60.De arXiv:1104.3515v3 [cond-mat.stat-mech] 23 Nov 2011 ∗ [email protected] † [email protected] ‡ [email protected] 1 I. INTRODUCTION Quenched disorder is known to have a pronounced effect on phase transitions in low dimensional systems. As Imry and Ma pointed out in 1975 [1], disorder can prevent the appearance of discontinuities, and of long range order, associated with first order transitions. Nonetheless, for a number of years there was considerable uncertainty about the mechanisms involved in such a “rounding effect”, and consequently also of the circumstances in which this effect appears [2].
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The Ising Model in Three and Four Dimensions
Scientific project: The Ising Model in three and four dimensions Michael Aizenman Hugo Duminil-Copin Princeton University University of Geneva Goal of the proposal The proposed collaboration will connect a leading young researcher, of outstanding promise in his cohort, with an established au- thority in the field. Thematically, the collaboration would link Modern Proba- bility with related Mathematical Physics from which many of its deep challenges originate. It is expected to also form a basis for a network which has room for growth, seeding a collaboration on which a future European Research Council (ERC) grant application (by Hugo Duminil-Copin) may be based for a collabo- rative program which may grow to involve students and other young researchers, and possibly also other institutions. Princeton and Geneva Universities possess exceptionally strong groups in the respective fields and the collaboration be- tween the two principal investigators could extend to other members of these two research groups in order to tackle related problems. Scientific motivation Since its formulation by Lenz [Len20], the Ising model has been the most studied example of a system undergoing a phase transition. The model was intended to explain the fact, discovered by Pierre Curie in 1895, that a ferromagnet loses its magnetization when heated above a critical temper- ature. Beyond its original motivation, the Ising model and its phase transition were found to be of a rather broad relevance, though of course its features do not exhaust the range of possible behaviors in statistical mechanics. The model also provided the testing ground for a large variety of techniques which have been later used to study other models.
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IAMP News Bulletin July 2016
IAMP News Bulletin July 2016 International Association of Mathematical Physics Contents International Association of Mathematical Physics News Bulletin, July 2016 Contents CMP in the Twenty-First Century3 Obituary: Roland Sénéor 10 Spectral Geometry Conference and the Legacy of Mark Kac 17 News from the IAMP Executive Committee 19 Contact Coordinates for this Issue 22 Bulletin Editor Editorial Board Evans Harrell Rafael Benguria, Virginie Bonnaillie-Noël, Yasuyuki Kawahigashi, Manfred Salmhofer, Robert Sims Contacts. http://www.iamp.org and e-mail: [email protected] Cover picture: Mathematical physics conference at Rome, 1977. See the article by Michael Aizenman in this issue. The views expressed in this IAMP News Bulletin are those of the authors and do not necessarily represent those of the MIAMP Executive Committee, Editor or Editorial Board. Any complete or partialΦ performance or reproduction made without the consent of the author or of his successors in title or assigns shall be unlawful. All reproduction rights are henceforth reserved, and mention of the IAMP News Bulletin is obligatory in the reference. (Art.L.122-4 of the Code of Intellectual Property). ISSN 2304-7348 News Bulletin (International Association of Mathematical Physics) 2 ∩IAMP News Bulletin, July 2016 CMP in the Twenty-First Century CMP in the Twenty-First Century by Michael Aizenman (Princeton University) Michael Aizenman is Thomas D. Jones Professor of Mathematical Physics at Princeton University. He served on the IAMP Executive Committee in 1997{1999, as Editor of Communications in Mathematical Physics from 1986 to 1993, and as Editor-in-Chief from 2001 to 2012. To every age its art.
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Complete List of Abstracts
Contents Entropy of Eigenfunctions ........................................ 1 Nalini Anantharaman, Herbert Koch and Stéphane Nonnenmacher 1 Motivations.............................................. 1 2 MainResult.............................................. 4 3 Outline of the Proof . 7 3.1 Deﬁnition of the Metric Entropy . 7 3.2 From Classical to Quantum Dynamical Entropy . 9 3.3 Entropic Uncertainty Principle. 13 3.4 Applying the Entropic Uncertainty Principle to the Laplacian Eigenstates . 14 References . 21 Stability of Doubly Warped Product Spacetimes ..................... 23 Lars Andersson 1 Introduction . 23 2 Warped Product Spacetimes . 24 2.1 AsymptoticBehavior .............................. 26 3 Fuchsian Method . 27 3.1 VelocityDominatedEquations....................... 28 3.2 VelocityDominatedSolution........................ 29 4 Stability . 30 References . 31 Rigorous Construction of Luttinger Liquids Through Ward Identities ... 33 Giuseppe Benfatto 1 Introduction . 33 2 The Tomonaga Model with Infrared Cutoff . 34 3 TheRGAnalysis.......................................... 35 4 TheDysonEquation....................................... 37 5 The First Ward Identity . 39 6 The Second Ward Identity . 40 xxvii xxviii Contents 7 The Euclidean Thirring Model . 41 References . 43 New Algebraic Aspects of Perturbative and Non-perturbative Quantum Field Theory ................................................... 45 Christoph Bergbauer and Dirk Kreimer 1 Introduction . 45 2 Lie and Hopf Algebras of Feynman Graphs . 46 3 From Hochschild Cohomology
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Variational Monte Carlo Renormalization Group
Variational Monte Carlo Renormalization Group Yantao Wu A Dissertation Presented to the Faculty of Princeton University in Candidacy for the Degree of Doctor of Philosophy Recommended for Acceptance by the Department of Physics Adviser: Roberto Car May 2021 © Copyright by Yantao Wu, 2021. All rights reserved. Abstract The renormalization group is an important method to understand the critical behaviors of a statistical system. In this thesis, we develop a stochastic method to perform the renormalization group calculations non-perturbatively with Monte Carlo simulations. The method is variational in nature, and involves minimizing a convex functional of the renormalized Hamiltonians. The variational scheme overcomes critical slowing down, by means of a bias potential that renders the coarse-grained variables uncorrelated. When quenched disorder is present in the statistical system, the method gives access to the flow of the renormalized Hamiltonian distribution, from which one can compute the critical exponents if the correlations of the renormalized couplings retain finite range. The bias potential again reduces dramatically the Monte Carlo relaxation time in large disordered systems. With this method, we also demonstrate how to extract the higher-order geometrical information of the critical manifold of a system, such as its tangent space and curvature. The success of such computations attests to the existence and robustness of the renormalization group fixed-point Hamiltonians. In the end, we extend the method to continuous-time quantum Monte Carlo simulations, which allows an accurate determination of the sound velocity of the quantum system at criticality. In addition, a lattice energy-stress tensor emerges naturally, where the continuous imaginary-time direction serves as a ruler of the length scale of the system.
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