Spring 2014
Monday 4.15  5.15 pm
Room E17139
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 SomethingAbstract: 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 MarcenkoPastur 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 (twosided) 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 LeeYang theorem implies that that for nonnegative 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 Vlasovtype 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 socalled Liouville measure defined formally by "M(dx)=e^{γX}dx" (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 supercritical 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 E17133
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 logconcave 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 cutoff 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 wellknown, 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 (martingalerelated) 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 nonuniqueness of candidates for geodesic lines.
A key argument, that allows to find a stationary law for the glueing process, is the cutoff 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 systemspanning avalanches
Abstract: A sandpile on a graph is an integervalued 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 s_{0} 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 s_{0} goes to negative infinity. I will outline the proof of this conjecture in http://arxiv.org/abs/1402.3283 and explain the bigpicture motivation, which is to give more predictive power to the theory of "selforganized 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 Nonequilibrium Dynamics and Random Matrices

January 5April 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 nonequilibrium dynamics
Schools

April 2125, 2014
Minerva Lecture Series at Columbia, by Alexei Borodin

June 227, 2014
PIMS Summer School in Probability

June 2327, 2014
School and Workshop on Random Interacting Systems

July 718, 2014
MSRI Summer Graduate School on Stochastic Partial Differential Equations

July 14August 1, 2014
ICTP Summer College on Nonlinear Dynamics, Instabilities and Patterns in Classical and Quantum Systems

July 2126, 2014
Heidelberg University Summer School on Mathematical Physics, Analysis and Stochastics

Janurary 59, 2015
Introductory school at CIRM (Marseille, France) : Disordered systems, random spatial processes and some applications
Conferences and Lecture Series

April 11, 2014
ColumbiaPrinceton Probability Day, Columbia

April 2325, 2014
Probability, Analysis and Dynamics, Bristol

May 1820, 2014
Frontier Probability Days, University of Arizona

May 1924, 2014
KPZ week, Toulouse, France

June 1620, 2014
NSF/CBMS Conference on Quantum Spin Systems, University of Alabama at Birmingham

July 28August 1, 2014
37th Conference on Stochastic Processes and their Applications, Buenos Aires, Argentina

August 611, 2014
7th International Conference on Stochastic Analysis and its Applications, Seoul, Korea

August 1321, 2014
International Congress of Mathematicians, Seoul, Korea

August 2226, 2014
International Conference on Quantum Probability and Related Topics, Seoul, Korea
Spring 2014 Organizers
 Alexei Borodin
 Ivan Corwin
 Vadim Gorin
 Alice Guionnet
 Jason Miller
 Scott Sheffield
 Charles Smart
 Omer Tamuz
 Hao Wu