Schedule

• February 10

  Arthur Jaffe (Harvard)

  On reflection positivity

  Abstract: We review how reflection positivity plays a fundamental role in the relation between probability and quantum theory. Also we describe the two new ways that it occurs: in the study of complex functionals and the study of Majoranas.

• February 17

  Presidents' day

• February 24

  Alan Edelman (MIT)

  Hermite, Laguerre and Jacobi
  Listen to Random Matrix Theory:It's Trying to Tell Us Something

  Abstract: In orthogonal polynomial theory one quickly learns about the special place occupied by Hermite, Laguerre, and Jacobi polynomials. I have often wondered why they are so special. I have heard dozens of correct answers. I am still not satisfied that I really know. Somehow in probability, Hermite, Laguerre, and Jacobi takes the form of Gaussian, Chi, and Beta Random variables. In computation, the triad takes the form of the symmetric eigenvalue problem, the singular value decomposition, and the lesser known, but very useful, generalized singular value decomposition. In Random Matrix Theory we have the Gaussian ensembles, the Wishart Matrices, and the Manova Matrices. Their limits are the Semicircle Law, the Marcenko-Pastur Law, and the Wachter Law. In combinatorics, we have the Catalan Numbers, the Narayana Trinagle, and a pyramid. This talk, intended to be accessible to a wide audience, will illustrate these ideas from a random matrix theory viewpoint. Discuss multivariate orthogonal polynomials, and the random matrix method of "Ghosts and Shadows."

• March 3

  Charles Newman (NYU)

  Statistical Mechanics and the Riemann Hypothesis

  Abstract: In this talk we review a number of old results concerning certain statistical mechanics models and their possible connections to the Riemann Hypothesis.

  A standard reformulation of the Riemann Hypothesis (RH) is: The (two-sided) Laplace transform of a certain specific function Psi on the real line is automatically an entire function on the complex plane; the RH is equivalent to this transform having only pure imaginary zeros. Also Psi is a positive integrable function, so(modulo a multiplicative constant C) is a probability density function.

  A (finite) Ising model is a specific type of probability measure P on the points S=(S_1,...,S_N) with each S_j = +1 or -1. The Lee-Yang theorem implies that that for non-negative a_1, ..., a_N, the Laplace transform of the induced probability distribution of a_1 S_1 + ... + a_N S_N has only pure imaginary zeros.

  The big question here is whether it's possible to find a sequence of Ising models so that the limit as N tends to infinity of such distributions has density exactly C Psi. We'll discuss some hints as to how one might try to do this.

• March 10

  Brian Rider (Temple University)

  Universality for the Stochastic Airy Operator

  Abstract: The Stochastic Airy Operator first arose as the continuum limit of certain (generalizations of) ensembles of symmetric Gaussian random matrices in the vicinity of their spectral edge. We show that this picture persists for the general logarithmic gas on the line with uniformly convex polynomial potential. Based on joint work with Manjunath Krishnapur and Balint Virag.

• March 17

  Yuri Bakhtin (NYU)

  Burgers equation with random forcing in noncompact setting

  Abstract: The Burgers equation is one of the basic nonlinear evolutionary PDEs. The study of ergodic properties of the Burgers equation with random forcing began in 1990's. The natural approach is based on the analysis of optimal paths in the random landscape generated by the random force potential. For a long time only compact cases of the Burgers dynamics on a circle or bounded interval were understood well. In this talk I will discuss the Burgers dynamics on the entire real line with no compactness or periodicity assumption on the random forcing. The main result is the description of the ergodic components and existence of a global attracting random solution in each component. The proof is based on ideas from the theory of first or last passage percolation. This is a joint work with Eric Cator and Kostya Khanin.

• March 24

  Spring Vacation

• March 31

  Oleksandr Kutovyi (MIT)

  Markov evolutions for interacting particle systems in continuum

  Abstract: We analyze an interacting particle system with a Markov evolution in continuum. The corresponding Vlasov-type scaling, which is based on a proper scaling of corresponding Markov generators and has an algorithmic realization in terms of related hierarchical chains of correlation functions equations is studied. The existence of rescaled and limiting evolutions of correlation functions as well as convergence to the limiting evolution are shown.

• April 7

  Christophe Garban (ENS Lyon)

  Liouville Brownian motion

  Abstract: Let X be a Gaussian Free Field (GFF) in two dimensions. I will introduce a Feller process (P_t^X) on the plane which, a.s. in the realization of the GFF X, preserves the so-called Liouville measure defined formally by "M(dx)=e^γXdx" (with γ < γ_c=2). The Liouville measure was popularized a few years ago by Duplantier and Sheffield in the context of Liouville Quantum gravity. I will discuss the construction and the properties of this Feller process called the Liouville Brownian motion as well as some recent progresses on the super-critical Liouville Brownian motion, i.e. when γ > γ_c=2. This is based on joint works with N. Berestycki, R. Rhodes, and V. Vargas.

• April 14

  Omer Tamuz (MSR / MIT)

  Majority Dynamics and the Period Two Property

  Abstract: A group of people connected by a social network each start with some opinion in {0,1}. They then proceed to repeatedly update their opinions by conforming to those of the majority of their neighbors. This model, which has been studied for a few decades now in various fields (cf. synchronous zero temperature Glauber dynamics), has the curious property that each person eventually either converges to a fixed opinion or else, from some point on, oscillates between the two possible opinions with period two. We will study this model on infinite graphs and random graphs, showing some old results, some new ones, and some nice open questions.

• April 21

  Patriots' day

• April 24

  Thursday, room E17-133

  Leonid Pastur (Kharkov)

  On the Law of Large Numbers and the Central Limit Theorem for Linear Eigenvalue Statistics of Sample Covariance Matrices with Dependent Entries

  Abstract: We consider sample covariance matrices whose data matrices have independent columns but dependent column components, notably columns with the log-concave distribution. We find conditions on the column distribution for a limiting Normalized Counting Measure of eigenvalues to exist (this proves to be the standard MP law) and for the validity of the Central Limit Theorem for a wide class of linear eigenvalue statistics and we comment on related topics of geometric asymptotic analysis.

• April 28th

  Philippe Biane (CNRS)

  Concavification of free entropy

  Abstract: I will give an overview of free entropy, which concerns how random matrices approximate traces on von Neumann algebras, and its use in operator algebra theory, as well as some new developments leading to the solution of the "additivity problem".

• May 5

  Victor Kleptsyn (UMR)

  Towards rigorous construction for random metrics : the cut-off process. (On a joint work with M. Khristoforov and M. Triestino)

  Abstract: One of the open problems in the domain of quantum gravity is the one of constructing a random metric on a manifold as a limit of a multiplicative cascade; if constructed for the case of a disc or of a sphere, it can be thought as the realization of "exp(DGFF) |dz|".

  Though this problem is well-known, there are very few rigorous known results. One of them is the work of Benjamini and Schramm for the multiplicative cascades on the interval, where the sequence of distances forms a martingale. The (martingale-related) convergence of measures (going back to the works of Kahane) is a key element in a work of Duplantier and Sheffield on the KPZ formula. Finally, the results of Le Gall and Miermont show that one can consider a random metric on the sphere as a limit of random planar maps.

  The main result of our work is the rigorous construction of a random metric via multiplicative cascades on hierarchical graphs (like Benjamini's eight graph, dihedral hierarchical lattice, etc.), this situation being both still accessible due to the graph structure, but already complicated due to the high non-uniqueness of candidates for geodesic lines.

  A key argument, that allows to find a stationary law for the glueing process, is the cut-off process: instead of looking for a critical renormalization value, we "stabilize" the process in the supercritical regime by adding a "shortcut", and then pass to the "diagonal" limit (the renormalization parameter tends to the critical value, and at the same time the influence of the shortcut tends to zero).

• May 12

  Lionel Levine (Cornell)

  Sandpiles and system-spanning avalanches

  Abstract: A sandpile on a graph is an integer-valued function on the vertices. It evolves according to local moves called topplings. Some sandpiles stabilize after a finite number of topplings, while others topple forever. For any sandpile s₀ if we repeatedly add a grain of sand at an independent random vertex, we eventually reach a sandpile s_τ that topples forever. Statistical physicists Poghosyan, Poghosyan, Priezzhev and Ruelle conjectured a precise value for the expected amount of sand in this "threshold state" s_τ; in the limit as s₀ goes to negative infinity. I will outline the proof of this conjecture in http://arxiv.org/abs/1402.3283 and explain the big-picture motivation, which is to give more predictive power to the theory of "self-organized criticality".

Semester/Year Programs

• January - May 2014

  CIMI trimester on Partial differential equations and probability, Toulouse, France

• September 2013 - June 2014

  Institute for Advanced Studied Year Program on Non-equilibrium Dynamics and Random Matrices

• January 5-April 3, 2015

  IHP trimester on Disordered Systems, Random Spatial Processes and their Applications, Paris, France

  This trimester will include the following workshops:

  26 January 2015 - 30 January 2015

  Statistical physics methods in social and economic systems

  16 February 2015 - 20 February 2015

  Spin glasses, random graphs and percolation

  09 March 2015 - 13 March 2015

  Interacting particles systems and non-equilibrium dynamics

Schools

• April 21-25, 2014

  Minerva Lecture Series at Columbia, by Alexei Borodin

• June 2-27, 2014

  PIMS Summer School in Probability

• June 23-27, 2014

  School and Workshop on Random Interacting Systems

• July 7-18, 2014

  MSRI Summer Graduate School on Stochastic Partial Differential Equations

• July 14-August 1, 2014

  ICTP Summer College on Non-linear Dynamics, Instabilities and Patterns in Classical and Quantum Systems

• July 21-26, 2014

  Heidelberg University Summer School on Mathematical Physics, Analysis and Stochastics

• Janurary 5-9, 2015

  Introductory school at CIRM (Marseille, France) : Disordered systems, random spatial processes and some applications

Conferences and Lecture Series

• April 11, 2014

  Columbia-Princeton Probability Day, Columbia

• April 23-25, 2014

  Probability, Analysis and Dynamics, Bristol

• May 18-20, 2014

  Frontier Probability Days, University of Arizona

• May 19-24, 2014

  KPZ week, Toulouse, France

• June 16-20, 2014

  NSF/CBMS Conference on Quantum Spin Systems, University of Alabama at Birmingham

• July 28-August 1, 2014

  37th Conference on Stochastic Processes and their Applications, Buenos Aires, Argentina

• August 6-11, 2014

  7th International Conference on Stochastic Analysis and its Applications, Seoul, Korea

• August 13-21, 2014

  International Congress of Mathematicians, Seoul, Korea

• August 22-26, 2014

  International Conference on Quantum Probability and Related Topics, Seoul, Korea