Todd Kemp (UC San Diego)
Universality and Concentration in Random Matrices with Correlated Entries
Random matrix theory began with the study, by Wigner in the 1950s, of high-dimensional matrices with i.i.d. entries (up to symmetry). The empirical law of eigenvalues demonstrates two key phenomena: universality (the limit empirical law of eigenvalues doesn't depend on the laws of the entries) and concentration (the convergence is robust and fast).
Several papers over the last decade (initiated by Bryc, Dembo, and Jiang in 2006) have studied certain special random matrix ensembles with structured correlations between some entries. The limit laws are different from the Wigner i.i.d. case, but each of these models still demonstrates universality and concentration.
In this lecture, I will talk about very recent results of mine and my students on these general phenomena:
Universality holds true whenever there are constant-width independent bands, regardless of the correlations within each band. (Interestingly, the same is not true for independent rows or columns, where universality fails.) I will show several examples of such correlated band matrices generalizing earlier known special cases, demonstrating how the empirical law of eigenvalues depends on the structure of the correlations.
At the same time, I will show that concentration is a more general phenomenon, depending not on the the structure of the correlations but only on the sizes of correlated partition blocks. Under some regularity assumptions, we find that Gaussian concentration occurs in NxN ensembles so long as the correlated blocks have size smaller than N^2/log(N).
Nicholas Simm (Warwick)
Subcritical multiplicative chaos and Gaussian field convergence in random matrix theory
Abstract. The height functions of a wide class of models in combinatorics, representation theory, or random matrices are known to converge to a Gaussian free field (GFF), or perhaps certain one dimensional slices of the GFF. In this talk I will discuss the question of exponentiating this statement, for certain toy models arising in random matrix theory. This question is not apriori straightforward because the GFF is a distribution rather than a function, so performing non-linear operations on it requires certain renormalization procedures. The exponential of the GFF has particularly important applications in topics like Liouville quantum gravity. Our results show that, after smoothing on some small scales, the renormalization can be made directly from a random matrix model. This is joint work with Gaultier Lambert (Zurich) and Dmitry Ostrovsky.
September 25 3:15--4:15
Douglas Rizzolo (University of Delaware)
From Markov on chains on sets of trees to diffusions on the space of interval partitions
Abstract. In 1999 David Aldous conjectured the existence of a diffusive limit for the natural ``simple random walk'' on the set of binary trees. In this talk we will discuss recent work on intertwining relations for this Markov chain and how they give insight into the conjectured limiting process. In particular, we will see how they suggest introducing certain diffusions on the space of interval partitions with Poisson-Dirichlet stationary distributions.
We will discuss how to construct these diffusions, which turn out to be part of a family of diffusions related to Petrov's two-parameter extension of Ethier and Kurtz's infinitely-many-neutral-alleles diffusion model. Based on joint work with Noah Forman, Soumik Pal, and Matthias Winkel.
September 28 3:00 -- 4:00 , Room 2-142,
Herbert Spohn (Techical University Munich and Columbia University)
KPZ growing interfaces: How flat is flat?
Abstract. Studied is the TASEP with initial data given through a spatially stationary stochastic
point process at density 1/2. For the height above the origin a novel one-parameter
family of asymptotic universal distributions is obtained, which interpolates between
GOE Tracy-Widom and Baik-Rains. We also discuss the half-space six-vertex model
at its conical point. This is joint work with S. Chitta and P.L. Ferrari.
Charles River Lectures
October 16 ROOM 2-147,
3:15 -- 4:15 Sasha Sodin (Queen Mary University of London)
The logarithmic derivative of the zeta-function on the critical line
Assuming the Riemann hypothesis, the logarithmic derivative of the zeta function is Cauchy distributed on the critical line. We shall discuss the derivation of this fact and its relation to a general result of Aizenman and Warzel.
4:15-5:15 Yinon Spinka (Tel-Aviv University)
A condition for long-range order in discrete spin systems
We present a new condition for the existence of long-range order in discrete spin systems, which emphasizes the role of entropy and high dimension. The condition applies to all symmetric nearest-neighbor discrete spin systems with an internal symmetry of `dominant phases'. Specific applications include a proof of Koteckýnjecture (1985) on anti-ferromagnetic Potts models, a strengthening of results of Lebowitz-Gallavotti (1971) and Runnels-Lebowitz (1975) on Widom-Rowlinson models and of Burton-Steif (1994) on shifts of finite type, and a significant extension of results of Engbers-Galvin (2012) on random graph homomorphisms on the hypercube.
No background in statistical physics will be assumed and all terms will be explained thoroughly. Joint work with Ron Peled.
Nathan Ross (University of Melbourne)
Comparing exponential and Erdos-Renyi random graphs, and a general bound on the distance between Bernoulli random vectors
Abstract. We present a general bound on the Wasserstein distance between two arbitrary distributions of sequences of Bernoulli random variables. The bound is in terms of 1) a mixing quantity for the Glauber dynamics of one of the sequences, and 2) a simple expectation of the other. The result is applied to estimate, with explicit error, expectations of functions of random vectors for some Ising models and exponential random graphs in "high temperature" regimes. Joint work with Gesine Reinert.
David Brydges (UBC)
Surface representations in lattice gauge theory
I will review
1) Wilson's definition for lattice gauge theory with compact gauge
2) Wilson loops and their physical interpretation,
3) derivation of random walk representations for the massless free
field and the Dynkin isomorphism,
4) the extension of (3) to surface representations for Wilson loop
expectations in cases G=Z_2, U(1) and SU(2).
Dmitry Dolgopyat (University of Maryland)
Local Limit Theorem for Nonstationary Markov chains
Dobrushin and Sethuuraman-Varadhan have proved sharp Central Limit
Theorem for additive functionals of finite non-stationary Markov chains.
We discuss the Local Limit Theorem in the same setting and give some
extensions and applications. Joint work with Omri Sarig.
Rick Kenyon (Brown)
Limit shapes beyond the complex Burgers equation
Abstract. We discuss the "5 vertex model", a model of random discrete interfaces. Limit shapes
in the model are solutions of a PDE generalizing the complex Burgers equation.
We show that solutions to this equation can be parameterized with analytic functions.
Peter Winkler (Dartmouth)
The Probability Puzzle that Spawned 100 Philosophy Papers
Proposed 17 years ago by philosopher Adam Elga, "Sleeping Beauty"
seems to be a simple question about probability. But there are some
who believe it exposes a gap---or, worse, a contradiction---in the
foundations of probability.
We'll describe the various "camps" in the controversy, and critique
their arguments from a probabilist's perspective. In the end, you'll
have to decide for yourself what the answer to the puzzle is---and, if
necessary, make peace with your own intuition.
Kay Kirkpatrick (Urbana-Champaign)
Two Quantum Central Limit Theorems
We will discuss a quantum central limit theorem in the context of Bose-Einstein condensation (BEC), a special phase of matter near absolute zero in which a gas of quantum particles can condense and behave as if it were a single giant quantum particle.
This work also makes the rigorous connection between the physics of the microscopic many-body dynamics and the mathematics of the macroscopic PDEs often used as reduced descriptions of BEC.
We will also discuss a central limit theorem for generators of quantum groups: that the joint distributions with respect to the Haar state of the generators of free orthogonal quantum groups converge to free families of generalized circular elements in the large (quantum) dimension limit. There is also a connection to free Araki-Woods factors. (Joint work with Gerard Ben Arous, Michael Brannan, Benjamin Schlein, and Gigliola Staffilani.)
Yash Deshpande (MIT)
An information-theoretic analysis of the stochastic block model
The stochastic block model is a popular model for large networks with latent clustered structure. Over the last few years, there has been significant interest in the statistical, computational and robustness aspects of the stochastic block model. In particular, insights from statistical physics have been particularly instrumental in spurring this interest and providing sharp predictions via non-rigorous methods. In this talk, we will consider information-theoretic view of the simplest stochastic block model and rigorously establish much of the picture provided by statistical physics, in a certain "diverging degree" limit. We will also show how information-theoretic quantities are intimately related with fundamental limits in estimation.
The analysis proceeds by a universality argument reducing the stochastic block models to (possibly more) familiar low-rank perturbations of GOE matrices. This is established through the classical Lindeberg swapping trick. Thanks to the Gaussian structure, the low-rank perturbed models are then amenable to a variety of analysis methods. We will take an algorithmic route using a particular "approximate message passing" scheme to establish the final result.
Joint work with Emmanuel Abbe and Andrea Montanari.
Remi Rhodes (Paris-Est)
Polyakov's formulation of 2d bosonic string theory
Mathematical construction et convergence of partition function of string theory is a longstanding problem born in 80's. In this talk we will present a mathematical/probabilistic construction of 2d bosonic string (also called noncritical strings). The framework we adopt was proposed by Polyakov in his seminal paper "Quantum geometry of bosonic strings". It describes a functional integral over the set of metrics on a Riemann surface with fixed topology. An essential feature of our approach is that it is probabilistic and non perturbative. The interest of our result is twofold. First, to the best of our knowledge, this is the first mathematical result about convergence of string theories. Second, our construction describes conjecturally the scaling limit of higher genus random planar maps weighted by the discrete Gaussian Free Field.
Vincent Vargas (ENS)
Liouville theory and the DOZZ formula
Liouville conformal field theory (LCFT hereafter), introduced by Polyakov
in his 1981 seminal work "Quantum geometry of bosonic strings", can be
seen as a random version of the theory of Riemann surfaces. A major issue
in theoretical physics was to solve the theory, namely compute the
correlation functions. In this direction, an intriguing formula for the
three point correlations of LCFT was proposed in the middle of the 90's by
Dorn-Otto and Zamolodchikov-Zamolodchikov, the celebrated DOZZ formula.
Recently, we gave a rigorous probabilistic construction of Polyakov's path
integral formulation of LCFT using the Gaussian Free Field. In this talk,
I will show that the three point correlation functions of the
probabilistic construction indeed satisfy the DOZZ formula. This
establishes an explicit link between probability theory (or statistical
physics) and the so-called conformal bootstrap approach of LCFT. Based on
a series of joint works with David, Kupiainen and Rhodes.