P R O B A B I L I T Y    S E M I N A R,    2 0 1 2 - 2 0 1 3

Organized with H.-T. Yau, room 232 in the Science Center, Harvard, on Thursdays 3-4pm.
Click on the title to read the abstract.

Spring Semester 2013

• Jan. 25, Gaetan Borot.   All-order asymptotic of beta ensembles in the multi-cut regime.
  The asymptotic behavior for the partition function and various statistics in the one hermitian matrix model has been intensively studied in the past decades. When the eigenvalues condensate on a single segment of the real line, one expects asymptotic expansions in powers of 1/N^2, where N denotes the size of the matrix. Physicists know this under the name of "topological expansion" since the 70s, and have developed algebro-geometric tools to compute them, culminating in the topological recursion of Eynard and Orantin. From the probabilistic point of view, the existence of such an expansion has been established for off-critical, real-analytic potential by Albeverio, Pastur, Shcherbina (and we will assume those conditions as well).
          The asymptotic behavior is different when the eigenvalues condensate on several disjoint segments (the so-called multi-cut regime), due to possible tunneling of eigenvalues between the segments. Interferences appear, and most quantities are now expected to have an expansion in 1/N, whose coefficients themselves are bounded functions of N with fast oscillations, given by Theta functions and their derivatives. The heuristic derivation of this behavior has been proposed by Bonnet, David and Eynard.
          In a joint work with Alice Guionnet, we show how to prove such asymptotic expansions to all orders, by combining rough probabilistic estimates with the analysis of Schwinger-Dyson equations. As a first consequence, we can obtain the all-order asymptotic expansion of orthogonal polynomials outside of the cuts, and of their recurrence coefficients. Our methods provides an alternative to the Riemann-Hilbert techniques to study the leading order, allow more easily to tackle the subleading orders. They also have a wider range of applicability, since they work same in all beta ensembles, where eigenvalues repel each other like the \beta-power of the distance, and the values \beta = 1,2,4 related to integrability have nothing special to this respect.


• Feb. 7, Jason Miller.   Imaginary Geometry and the Gaussian Free Field.
  The Schramm-Loewner evolution (SLE) is the canonical model of a non-crossing conformally invariant random curve, introduced by Oded Schramm in 1999 as a candidate for the scaling limit of loop erased random walk and the interfaces in critical percolation. The development of SLE has been an exciting area in probability theory over the last decade because Schramm’s curves have now been shown to arise as the scaling limit of the interfaces of a number of different discrete models from statistical physics. In this talk, I will describe how SLE curves can be realized as the flow lines of a random vector field generated by the Gaussian free field, the two-time-dimensional analog of Brownian motion, and how this perspective can be used to study the sample path behavior of SLE. This talk is based on joint work with Scott Sheffield.


• Feb. 14, Michael Brenner.   Signal identification from sample covariance matrices with correlated noise.
  We find that the spectral characteristics of sample covariance matrices derived from biological datasets (microarrays, RNASeq) are well described by a model in which a signal is corrupted by correlated noise. Such eigenvalue distributions have long tails, and the signal and noise contributions mix with each other in the tails. A mathematical analysis demonstrates that information about the signal can be recovered by finding correlations between large magnitude components of the eigenvectors of the sample covariance matrix. The alignment of such eigenvectors with the signal most likely occurs when the corresponding eigenvalues are close to the signal eigenvalue, with the strength of the alignment being controlled by the ratio of the eigenvalue spacing to the size of typical elements in the sample covariance. Within a long tail of the eigenvalue distribution, the eigenvalue spacing is atypically large, making strong alignment possible.


• Feb. 21, Benedek Valkó.   Diffusivity of lattice gases.
  We consider one component lattice gases with a local dynamics and a stationary product Bernoulli measure. We give upper and lower bounds on the diffusivity at an equilibrium point depending on the dimension and the local behavior of the macroscopic flux function. We show that if the model is expected to be diffusive, it is indeed diffusive, and, if it is expected to be superdiffusive, it is indeed superdiffusive. (Joint with J. Quastel.)


• Feb. 28, Nalini Anantharaman.   Quantum Ergodicity on Large Regular Graphs.
  ``Quantum ergodicity'' usually deals with the study of eigenfunctions of the Laplacian on Riemannian manifolds, in the high-frequency asymptotics. The rough idea is that, under certain geometric assumptions (like negative curvature), the eigenfunctions should become spatially uniformly distributed, in the high-frequency limit. I will review the many conjectures in the subject, some of which have been turned into theorems recently. Physicists like Uzy Smilansky or John Keating have suggested looking for similar questions and results on large (finite) discrete graphs. Take a large graph $G=(V, E)$ and an eigenfunction $\psi$ of the discrete Laplacian -- normalized in $L^2(V)$. What can we say about the probability measure $|\psi(x)|^2$ ($x\in V$)? Is it close to uniform, or can it, on the contrary, be concentrated in small sets? I will talk about ongoing work with Etienne Le Masson, in the case of large regular graphs.


• Mar. 7, Ivan Corwin.   From duality to determinants for ASEP.
  I will explain duality for the asymmetric simple exclusion process and then solve the associated quantum many body problem for special initial conditions via a contour integral ansatz. Using the resulting formulas I develop generating series which identify particle location distributions and which can be expressed as Fredholm determinants (one of which coincides with the Tracy-Widom ASEP formula). Asymptotics of such expressions are readily computable and lead to Kardar-Parisi-Zhang universality class fluctuations. In essence, this is a rigorous discrete deformation of the physicist's (famously non-rigorous) polymer replica trick. This is based on joint work with Alexei Borodin and Tomohiro Sasamoto.


• Mar 14, Benoit Collins.   Norm convergence for random matrices and applications.
  We will review some recent results about the convergence in the large N limit of the operator norm of random matrix models obtained as non-commutative polynomials in i.i.d unitary Haar distributed matrices. We will describe some applications of these results, in particular to the study of singular values of random subspaces of a tensor product. This talk is based on joint works with S. Belinschi, M. Fukuda, C. Male, I. Nechita.


• Mar 26, Noureddine El Karoui.   Some connections between random matrix theory and high-dimensional statistics.
  With data collection and storage easier by the day, statisticians are now working with datasets having many observations and many measurements per observations. This is a radical shift from the classical theory of statistics, where the number of observations (n) is most often assumed to be much bigger than the number of measurements per observations (p). It is therefore important to understand how statistical estimators behave and perform in the setting where both p and n are large. Because random matrices are a key building block of statistics, the theory of high-dimensional random matrices is helpful in shedding light on certain high-dimensional (i.e large p, large n) statistical problems. In this talk, I will discuss some of these problems, as well as potential limitations of the standard models of random matrix theory when used in a statistical setting.


• Apr. 4, Jonathan Novak.   An invitation to Weingarten calculus.
  The integration of polynomial functions on compact matrix groups is a challenging problem which has a long history in mathematical physics, with various partial results scattered throughout the literature. In recent years, the needs of random matrix theory have led to the development of a structured approach to this problem which has come to be known as "Weingarten calculus." This calculus is analogous to Wick-type methods for computing averages of polynomial functions in Gaussian random variables, but it is more involved both combinatorially and algebraically. I'll describe the modern formulation of Weingarten calculus and, if time permits, explain how combining Weingarten calculus and Schwinger-Dyson equations leads to asymptotics for Schur polynomials. This talk is partly based on a book in preparation with Benoit Collins and Sho Matsumoto, and ongoing work with Alice Guionnet.


• Apr. 11, Vadim Gorin.   2-D structures of beta-random matrix ensembles and the Gaussian Free Field.
  I will present an approach for studying classical beta-random matrix ensembles through (the limits of) Macdonald polynomials and Macdonald difference operators. We will discuss several results along these lines. The main result concerns global fluctuations of multilevel beta-Jacobi ensemble and its degenerations. I will explain that these fluctuations are asymptotically governed by the Gaussian Free Field - 2-D analogue of the Brownian Motion.


• Apr. 18, Mihai Stoiciu.   Transition in the Microscopic Eigenvalue Distribution for Random and Deterministic Unitary Operators.
  We consider several classes of random and deterministic unitary operators and investigate their microscopic eigenvalue distribution. We show that these operators exhibit a transition in their microscopic eigenvalue distribution, depending on the properties of the corresponding spectral measures (we investigate random measures and measures associated to hyperbolic reflection groups). In the case of pure point spectral measures, the microscopic eigenvalue distribution is Poisson (no correlation). As the spectral measures approach an absolutely continuous measure, the repulsion between the eigenvalues increases and the microscopic eigenvalue distribution converges to the clock (or "picket fence") distribution.


• Apr. 25, Nikolai Makarov.  Random normal matrices and Ward identities.
  I will review the method of Ward identities in the context of random normal matrix model. In particular, I will focus on the distribution of eigenvalues near the boundary of the support of the equilibrium measure. (Joint work with Y. Ameur and H. Hedenmalm, and with Y. Ameur and N.-G. Kang)


• Apr. 26, Random Matrices, Random Geometry, a joint Harvard-MIT probability afternoon.


• May 2, Jean-François Le Gall.  The harmonic measure of critical Galton-Watson trees.
  We consider simple random walk on a critical Galton-Watson tree conditioned to have height greater than n. It is well known that the cardinality of the set of vertices of the tree at generation n is then of order n. We prove the existence of a constant b < 1 such that the hitting distribution of the generation n in the tree by random walk is concentrated with high probability on a set of cardinality approximately equal to n^b . In terms of the analogous continuous model, the dimension of harmonic measure on a level set of the tree is equal to b, whereas the dimension of any level set is equal to 1. The constant b is approximately equal to 0.78 and can be expressed in terms of the asymptotic distribution of the conductance of large critical Galton-Watson trees. This talk is based on joint work with Nicolas Curien.


• May 3, Dan Stroock, 1.30pm. Determinants of Green's Functions with Applications to Counting Spanning Trees and Other Things.
  

Fall Semester 2012

• Sept. 7, Alan Hammond.   The multi-line Airy process and its Brownian-Gibbs property. Consider a large number N of one-dimensional Brownian bridges define on the interval [-N,N] and conditioned by mutual avoidance. The curves become ordered under the conditioning, with the highest one reaching a mean value of 2N at the midpoint 0. A parabolic scaling in x and y about the point (0,2N) - dilating the y-direction by N^{1/3} and the x-direction by N^{1/3} - leads in a weak limit in high N to a process supported on infinitely many non-intersecting curves. This process appears as a scaling limit of the height profile of natural models for growing interfaces. In joint work with Ivan Corwin, we prove that this "multi-line Airy process" is supported on continuous curves, and that it satisfies a Gibbs resampling property which is a natural counterpart to a resampling property enjoyed by a finite system of Brownian bridges. This probabilistic means of analysing a scaling limit object in the KPZ universality class is a rather different tool to the exact solution approaches which are often used; it yields several new results, and furnishes the final element in a proof of Johansson that the normalized lateral fluctuation of the geodesic in geometric first passage percolation has a limiting law.


• Oct. 4, Julien Dubedat.   Double dimers and tau-functions. The double dimer model is a variation of the classical dimer model consisting in superimposing two independent dimer configurations (perfect matchings) on a graph, thus creating an ensemble of non-intersecting loops. Kenyon has recently introduced and studied ``anyonic" correlators for this model. We discuss the convergence (in the small mesh limit) of some of these correlators to the tau-functions appearing in the theory of isomonodromic (SU(2)) deformations.


• Oct. 11, Alessio Figalli.   Stability results for geometric and functional inequalities. Geometric and functional inequalities play a crucial role in several problems arising in the calculus of variations, partial differential equations, geometry, probability, etc. More recently, there has been a growing interest in studying the stability for such inequalities. The basic question one wants to address is the following: suppose we are given a functional inequality for which minimizers are known. Can we prove that if a function almost attains the equality then it is close (in some suitable sense) to one of the minimizers? The aim of this talk is to describe some ways to attack this kind of problems, and show some applications.


• Oct. 18, Charles Smart.   Apollonian structure in the Abelian sandpile. The Abelian sandpile is a deterministic version of family of probabilistic chip firing models on the integer lattice. It is well known in part because the stable sandpiles are often beautiful fractal images. The sandpile dynamics has a continuum scaling limit, captured by a certain elliptic partial differential equation, with a surprising algebraic structure. In particular, there is an Apollonian circle packing that controls the dynamics and explains the fractal images.


• Oct. 26, Louis-Pierre Arguin.   Poisson-Dirichlet statistics for the extremes of log-correlated Gaussian fields. Gaussian fields with logarithmically decaying correlations, such as branching Brownian motion and the 2D Gaussian free field, are conjectured to form a new universality class of extreme value statistics (notably in the work of Carpentier & Ledoussal and Fyodorov & Bouchaud). This class is the borderline case between the class of IID random variables (or REM model), and models where correlations start to affect the statistics. In this talk, I will report on the recent rigorous progress in describing the new features of this class. In particular, I will describe the emergence of Poisson-Dirichlet statistics of the Gibbs measure of a non-hierarchical log-correlated Gaussian field similar to the Gaussian free field. This is joint work with Olivier Zindy.


• Oct. 30, Jürg Fröhlich.   What is it that is confusing about Quantum Mechanics? jointly organized with the Physics Department. Here are the slides for this talk. I will sketch a general algebraic formalism unifying classical and quantum physics. Among other purposes, this formalism serves to highlight the fundamental differences between classical and quantum theories of Nature. I will then attempt to clarify what it is that quantum theories tell us about Nature - in other words, what kind of a theory of Nature quantum mechanics is. With a little luck, I will be able to identify what it is that people find confusing about quantum mechanics and then explain why there is no reason to be confused about this wonderful theory.


• Nov. 8, Elizabeth Meckes.   The spectra of powers of random unitary matrices. It has long been recognized that the eigenvalues of random unitary matrices look very different from i.i.d. points on the circle: they are very evenly spaced. As you start raising a random unitary matrix to powers, the eigenvalues start to clump together, and by the time you raise a random matrix in U(N) to the Nth power, the eigenvalues look exactly like i.i.d. points on the circle. There is a good reason for this: Rains showed that the distributions are exactly the same, and that there is a kind of smooth interpolation between the two extremes. In joint work with M. Meckes, we quantify this phenomenon by proving sharp non-asymptotic estimates on the means and tails of the L_p Wasserstein distances between spectral measures of powers of random unitary matrices and the uniform measure on the circle. The bounds also allow us to obtain sharp almost-sure convergence rates as the size of the matrix tends to infinity. Along the way, we needed the long-suspected fact that the full unitary group satisfies the same type of log-Sobolev inequality as the special unitary group; I will sketch the proof of this result, which should be of independent interest.


• Nov. 13, Craig A. Tracy.   Bethe Ansatz Methods in Stochastic Integrable Models.
  The asymmetric simple exclusion process (ASEP) is not of determinantal class, but it is integrable in the sense that Bethe Ansatz methods lead to an explicit formula for the transition probability for N-particle ASEP. We sketch the proof of this fact. Two variations of ASEP are multispecies ASEP (first, second, etc. class particles) and ASEP on the semi-infinite lattice {1, 2, . . .}. In both cases the standard Bethe Ansatz has to be modified in order to compute the transition probability of the N-particle system. We explain these modifications and indicate how the proof of the main result is changed. This is joint work with Harold Widom.


• Nov. 20, Alice Guionnet, About heavy tailed random matrices.


• Nov. 29, Tatyana Shcherbina, Universality of the local regime of characteristic polynomials of the 1D Gaussian band matrices.