Date: August 28 to September 1, 2017

Main Lecturer: Alice Guionnet (Lyon)

Location: Columbia Mathematics Department

Funding: NSF CBMS conference grant DMS-1642595 and the Minerva Lecture Series

Organizers: Ivan Corwin and Yi Sun

Conference Poster: Poster

Understanding the large dimension asymptotics of random matrices or related models such as random tilings has been a hot topic for the last twenty years within probability, mathematical physics, and statistical mechanics. Because such models are highly correlated, classical methods based on independent variables fail. Special cases have been studied in detail thanks to specific forms of the laws, such as determinantal laws. These lectures will investigate a general class of models using the so called Dyson-Schwinger equations or generalizations such as Nekrasov's equations. The idea is similar to Stein's method in that the observables are approximate solutions of equations that can be solved asymptotically.

Alice Guionnet (Lyon) will give ten main lectures, divided into two per day. Besides these lectures there will be supplementary lectures by other senior researchers attending the school, including:

- Gaetan Borot (Bonn)

**Title:**1D discrete log gas in the macroscopic regime

**Abstract:**We will explain how fluctuations of linear statistics and asymptotic expansion of the partition function and correlation function of a 1d discrete log-gas can be carried out, in full analogy with the (continuous) 1d log gas/beta-ensembles. This discrete model is realized in 2d random lozenge tilings. In particular, we will describe the replacement for Selberg integral (with Jack binomial model) and of Dyson-Schwinger equations (with Nekrasov equations) which are instrumental in studying this discrete model. The talk is based on joint ongoing work with Gorin and Guionnet. - Paul Bourgade (NYU)

**Title:**Ward identities, the 2D Coulomb gas and the Gaussian free field

**Abstract:**We prove a quantitative central limit theorem for linear statistics of particles in the complex plane, with Coulomb interaction at any temperature. This generalizes works by Rider, Virag, Ameur, Hendenmalm and Makarov obtained for the inverse temperature beta=2. The main tools are a multi scale analysis and the Ward identity (or loop equation). This is joint work with Roland Bauerschmidt, Miika Nikula and Horng-Tzer Yau. - Vadim Gorin (MIT)

**Title:**Gaussian Free Field in random tilings

**Abstract:**The fluctuations of the height function of random lozenge and domino tilings of polygonal domains are believed to be governed by the 2d Gaussian Free Field in an appropriate complex structure. This was conjectured by Kenyon and Okounkov and they also suggested a neat geometric procedure for defining the corresponding complex structure.

I will present a new approach to this conjecture for a class of domains through gluings of Gelfand-Tsetlin patterns. It combines discrete Dyson-Schwinger equations with the method of Schur generating functions to yield the result. - Sylvia Serfaty (NYU)

**Title:**LDP and CLT for Log and Coulomb Gases

**Abstract:**We present a Large Deviation Principle for large systems of particles with logarithmic interactions in 1D and 2D, or more general inverse power (= Riesz) interactions, including Coulomb interactions. The LDP lies at next to leading order and is expressed in terms of the microscopic point processes or "empirical fields". In the case of the 1D and 2D logarithmic interactions, we also present a Central Limit Theorem for the fluctuations from the macroscopic distribution. All the results are valid for arbitrary values of the inverse temperature, and fairly general confining potentials. This is based on joint work with Thomas Leblé, and Florent Bekerman and Thomas Leblé.

Lecture | Description |
---|---|

Lectures 1 + 2 | Introduction to Dyson-Schwinger equations. Topological expansion for the GUE. |

Lectures 3 + 4 | CLT for beta model in the one cut case. |

Lectures 5 + 6 | CLT for lozenge tiling and discrete beta models. |

Lectures 7 + 8 | Topological expansions for large N asymptotics of partition functions. |

Lectures 9 + 10 | Generalization to several cut models. |

**Monday August 28**

Time | Description | Location |
---|---|---|

09:30 am - 10:00 am | Breakfast | Room 508 |

10:00 am - 11:00 am | Lecture 1 (Alice Guionnet) | Room 312 |

11:00 am - 11:30 am | Break | Room 508 |

11:30 am - 12:30 pm | Participant Talks Session 1 | Room 312 |

12:30 pm - 02:00 pm | Lunch | Room 508 |

02:00 pm - 03:00 pm | Lecture 2 (Alice Guionnet) | Room 312 |

03:00 pm - 04:30 pm | Problem Session | Room 312 |

04:30 pm - 05:00 pm | Break | Room 508 |

05:00 pm - 06:00 pm | Poster Session (Titles and Abstracts) | Room 508 |

**Tuesday August 29**

Time | Description | Location |
---|---|---|

09:30 am - 10:00 am | Breakfast | Room 508 |

10:00 am - 11:00 am | Lecture 3 (Alice Guionnet) | Room 312 |

11:00 am - 11:30 am | Break | Room 508 |

11:30 am - 12:30 pm | Participant Talks Session 2 | Room 312 |

12:30 pm - 02:00 pm | Lunch | Room 508 |

02:00 pm - 03:00 pm | Lecture 4 (Alice Guionnet) | Room 312 |

03:00 pm - 04:30 pm | Problem Session | Room 312 |

04:30 pm - 05:00 pm | Break | Room 508 |

05:00 pm - 06:00 pm | Talk (Paul Bourgade) | Room 312 |

**Wednesday August 30**

Time | Description | Location |
---|---|---|

09:30 am - 10:00 am | Breakfast | Room 508 |

10:00 am - 11:00 am | Lecture 5 (Guionnet) | Room 312 |

11:00 am - 11:30 am | Break | Room 508 |

11:30 am - 12:30 pm | Talk (Sylvia Serfaty) | Room 312 |

12:30 pm - 02:00 pm | Lunch | Room 508 |

02:00 pm - 03:00 pm | Lecture 6 (Guionnet) | Room 312 |

03:00 pm - 04:30 pm | Problem Session | Room 312 |

04:30 pm - 05:00 pm | Break | Room 508 |

05:00 pm - 06:00 pm | Participant Talks Session 3 | Room 312 |

**Thursday August 31**

Time | Description | Location |
---|---|---|

09:30 am - 10:00 am | Breakfast | Room 508 |

10:00 am - 11:00 am | Lecture 7 (Alice Guionnet) | Room 312 |

11:00 am - 11:30 am | Break | Room 508 |

11:30 am - 12:30 pm | Participant Talks Session 4 | Room 312 |

12:30 pm - 02:00 pm | Lunch | Off-Campus |

02:00 pm - 03:00 pm | Lecture 8 (Alice Guionnet) | Room 312 |

03:00 pm - 04:30 pm | Problem Session | Room 312 |

04:30 pm - 05:00 pm | Break | Room 508 |

05:00 pm - 06:00 pm | Talk (Vadim Gorin) | Room 312 |

**Friday September 1**

Time | Description | Location |
---|---|---|

09:30 am - 10:00 am | Breakfast | Room 508 |

10:00 am - 11:00 am | Lecture 9 (Alice Guionnet) | Room 312 |

11:00 am - 11:30 am | Break | Room 508 |

11:30 am - 12:30 pm | Talk (Gaetan Borot) | Room 312 |

12:30 pm - 02:00 pm | Lunch | Room 508 |

02:00 pm - 03:00 pm | Lecture 10 (Alice Guionnet) | Room 312 |

03:00 pm - 04:30 pm | Problem Session | Room 312 |

04:30 pm - 05:00 pm | Break | Room 508 |

05:00 pm - 06:00 pm | Additional Discussion | Room 312 |

**Session 1 (Monday August 28 11:30am-12:30pm)**

Speaker | Title and Abstract |
---|---|

Indrajit Jana (UC Davis) | Limiting Spectral Distribution of Random Band MatricesWe consider the limiting spectral distribution of matrices of the form $\f{1}{2b_{n}+1} (R + X)(R + X)^{*}$, where $X$ is an $n\times n$ random band matrix of bandwidth $b_{n}$ and $R$ is a non random band matrix of bandwidth $b_{n}$. We show that the Stieltjes transform of spectrum of such matrices converges to the Stieltjes transform of a non-random measure. And the limiting Stieltjes transform satisfies an integral equation. For $R=0$, the integral equation yields the Stieltjes transform of the Marchenko-Pastur law. |

Tankut Can (CUNY) | Coulomb Plasma on a Singular SurfaceMotivated by the connection to the fractional quantum Hall (FQH) effect, we study the 2D Coulomb plasma on a surface with conical singularities. Employing a large N limit, we compute the dependence of the partition function on the complex structure moduli of the surface. The resulting variational formula is controlled by exact sum rules, which we find using a Ward identity (aka loop equation). Despite the fast decay of density correlation in the screening phase of the plasma, we find that conical singularities exhibit power law correlations, indicating the emergence of conformal symmetry. As conformal primaries, the conical singularities are characterized by a conformal dimension, which is fixed by a sum rule. In the FQH setting, the conformal dimension corresponds to an intrinsic angular momentum of a conical singularity, while the power law correlations encode the Berry phase picked up by the FQH wave function upon braiding singularities. |

Benson Au (UC Berkeley) | Traffic distributions of random band matricesWe study random band matrices within the framework of traffic probability, an operadic non-commutative probability theory introduced by Male based on graph operations. As a starting point, we revisit the familiar case of the permutation invariant Wigner matrices and compare the situation to the general case in the absence of this invariance. Here, we find a departure from the usual free probabilistic universality of the joint distribution of independent Wigner matrices. We then extend our analysis to random band matrices, as studied by Bogachev, Molchanov, and Pastur, and investigate the extent to which the joint traffic distribution of independent copies of these matrices deviates from the Wigner case. |

Ramis Movassagh (IBM) | Generic local Hamiltonians are gaplessWe prove that quantum local Hamiltonians with generic interactions are gapless. In fact, we prove that there is a \textit{continuous} \textit{density of states} arbitrary close to the ground state. The Hamiltonian can be on a lattice in any spatial dimension or on a graph with a bounded maximum vertex degree. The type of interactions allowed for herein include translational invariance in a disorder (i.e., probabilistic) sense with some assumptions on the local distributions. We calculate the scaling of the gap with the system's size in the case that the local terms are distributed according to gaussian β−orthogonal random matrix ensemble. As a corollary there exist finite size partitions with respect to which the ground state is arbitrarily close to a product state. In addition to the lack of an energy gap, we prove that the ground state is degenerate when the local eigenvalue distribution is discrete. This work excludes the important class of truly translationally invariant Hamiltonians where the local terms are all equal. |

**Session 2 (Tuesday August 29 11:30am-12:30pm)**

Speaker | Title and Abstract |
---|---|

Benjamin Landon (Harvard) | Local statistics of Dyson Brownian motionIn this talk we will present recent results on the local ergodicity of Dyson Brownian motion. In particular, convergence of the local statistics to that of the invariant GOE/GUE has been obtained for a wide class of initial data, in both the bulk and at the edge. We will also discuss applications of these results to random matrix theory, such as bulk universality for sparse random matrices and other ensembles going well beyond the class of Wigner matrices. Based on joint works with Z. Che, J. Huang, P. Sosoe and H.-T. Yau. |

Vivian Healey (UChicago) | The Loewner Equation with Branching and the Continuum Random TreeIn its most well-known form, the Loewner equation gives a correspondence between curves in the upper half-plane and continuous real functions (called the “driving function” for the equation). We consider the generalized Loewner equation, where the driving function has been replaced by a time-dependent real measure. We investigate the delicate relationship between the driving measure and the generated hull, specifying a class of discrete random driving measures that generate hulls in the upper half-plane that are embeddings of trees. We show that these driving measures have a scaling limit as the trees converge to the continuum random tree, and we describe links between this limiting superprocess, the complex Burgers equation, and Dyson Brownian motion. |

Brent Nelson (UCLA) | Dyson-Schwinger equations in free probabilityThe free probability analogue of Dyson–Schwinger equations have solutions that correspond to joint laws of non-commutative random variables. These equations have a potential function as a parameter, which is typically a formal power series in non-commuting variables satisfying some convergence condition. When a non-commutative Dyson–Schwinger equation has a potential function coming from a particular class of quadratic polynomials, the solution is known to be the joint law of semicircular operators that are either freely independent or "quasi-free" in the sense that they generate a free Araki–Woods factor. Starting with the incredible work of Guionnet and Shlyakhtenko in 2014, many examples of non-commutative (or "free") transport have been exhibited by studying non-commutative Dyson–Schwinger equations with potentials that are small perturbations of such quadratic potentials. In this talk, I will provide an introduction to these non-commutative notions and give a survey of free transport results. |

**Session 3 (Wednesday August 30 5:00pm-6:00pm)**

Speaker | Title and Abstract |
---|---|

Axel Saenz (Virginia) | Transition Probabilities for ASEP on the ringFor ASEP on the line, the system may never reach equilibrium dynamics depending on the initial conditions. Whereas for ASEP on the ring, one expects the system to reach equilibrium dynamics given enough time. In the special case of TASEP on the ring, there are recent result that give the specific crossover from KPZ dynamics and equilibrium dynamics. In collaboration with Z. Liu and D. Wang, we obtain the transition probability formulas for the periodic ASEP model. These formulas specialize to the formulas of ASEP on the line and TASEP on the ring, which are a first step to generalize the results of ASEP on the line and TASEP on the ring. |

Evgeni Dimitrov (MIT) | The ASEP and Hall-Littlewood Gibbsian line ensemblesWe consider the ASEP started from step initial condition and investigate the large time $T$ distribution of the height function. Conjecturally, under $T^{2/3}$ spatial and $T^{1/3}$ fluctuation scaling (also known as KPZ scaling) the asymptotic behavior is described by the Airy$_2$ process. We provide further evidence for this conjecture by showing that under the KPZ scaling the height function is tight in the space of continuous curves , and subsequential limits are absolutely continuous with respect to Brownian motion shifted by a parabola. This is based on joint work with Ivan Corwin. |

Pierre Yves Gaudreau Lamarre (Princeton) | The Stochastic Semigroup Approach to the Edge of Beta-EnsemblesA remarkable advance in the study of random matrices and associated point processes consists of the development of a theory of operator limits for such objects. First, I will briefly review some of the foundational results in this theory regarding Gaussian beta-ensembles and the stochastic Airy operator. Then, I will discuss more recent developments, including the identification of Feynman-Kac formulas for the stochastic Airy operator as well as novel connections between beta-ensembles and stochastic calculus. Based on work by Vadim Gorin and Mykhaylo Shkolnikov, and joint work with Mykhaylo Shkolnikov. |

**Session 4 (Thursday August 31 11:30am-12:30pm)**

Speaker | Title and Abstract |
---|---|

Izumi Okada (Tokyo) | High points in the range of simple random walks in two dimensionsWe consider the problem, as suggested by Dembo ($2003$, $2006$), of late points of a simple random walk in two dimensions. It has been shown that the exponents for the numbers of pairs of late points coincide with those of nearly favorite points and high points in the Gaussian free field, whose exact values are known. We estimate the exponents for the numbers of a multipoint set of late points, favorite points and high points in average and in probability. |

Roozbeh Gharakhloo (IUPUI) | On the asymptotic analysis of Toeplitz + Hankel determinantsWe want to analyse the asymptotics of a Toeplitz+Hankel determinant with Toeplitz symbol $\phi(z)$ and Hankel symbol $w(z)$. When symbols $\phi(z)$ and $w(z)$ are related in specific ways, the asymptotics of T+H determinants have been studied by E. Basor and T. Ehrhardt and by P. Deift, A. Its and I. Krasovsky. The distinguishing feature of this work is that we do not assume any relations between the symbols $\phi(z)$ and $w(z)$. In this talk, the Hankel symbol is a Jacobi weight and the Toeplitz symbol is assumed to be analytic in a neighborhood of the unit circle. we approach this problem by analysing a 4 by 4 Riemann Hilbert problem. This work is part of the joint research project with Alexander Its, Percy Deift and Igor Krasovsky. |

Mohammad Khabbazian (Columbia) | Novel Sampling Design for Respondent-driven SamplingRespondent-driven sampling (RDS) is a method of chain referral sampling popular for sampling hidden and/or marginalized populations. As such, even under the ideal sampling assumptions, the performance of RDS is restricted by the underlying social network: if the network is divided into communities that are weakly connected to each other, then RDS is likely to oversample one of these communities. In order to diminish the "referral bottlenecks" between communities, we propose anti-cluster RDS (AC-RDS), an adjustment to the standard RDS implementation. Using a standard model in the RDS literature, namely, a Markov process on the social network that is indexed by a tree, we construct and study the Markov transition matrix for AC-RDS. We show that if the underlying network is generated from the Stochastic Blockmodel with equal block sizes, then the transition matrix for AC-RDS has a larger spectral gap and consequently faster mixing properties than the standard random walk model for RDS. In addition, we show that AC-RDS reduces the covariance of the samples in the referral tree compared to the standard RDS and consequently leads to a smaller variance and design effect. We confirm the effectiveness of the new design using both the Add-Health networks and simulated networks. https://arxiv.org/pdf/1606.00387.pdf |

Presenter | Title and Abstract |
---|---|

Hyun-Jung Kim (USC) | Stochastic Anderson Spectrum of Wick TypeThe spectrum of stochastic Airy operator has been studied in the field of random matrix theory. It is well-known that the largest k eigenvalues of certain Hermitian matrices converge, in a suitable limit, to the smallest k eigenvalues of a stochastic Airy operator with usual product on the half-line. From Minami (2014), the stochastic Airy operator with usual product has the pure point spectrum with finite multiplicity in a pathwise sense. In a similar manner, Fukushima and Nakao (1977) proved that the stochastic Anderson operator with usual product has the pure point spectrum of finite multiplicity with no accumulation point except for +infinity. We now mainly focus on the eigenvalue problem of stochastic Anderson operator with Wick product. With the help of Malliavin calculus and Fredholm alternatives, we achieve a recurrence relation and patterns for the eigenvalues. Moreover, we investigate their related properties and find connections to the case of usual product. |

Andrei Prokhorov (IUPUI) | The smallest eigenvalue distribution of incomplete Laguerre unitary ensemble(joint work with Alexander Its and Thomas Bothner) We consider the Laguerre unitary ensemble. We perform the thinning procedure, removing each eigenvalue with the same probability $1-\gamma$. We study the limiting distribution of the smallest eigenvalue of the obtained ensemble. It is the hard edge of the spectrum and the behavior is described by the integrable Fredholm determinant with Bessel kernel. We perform the nonlinear steepest descent analysis of corresponding Riemann-Hilbert problem and deduce the asymptotics of the determinant. For evaluation of the constant term in the asymptotics we interpret the determinant as Jimbo-Miwa-Ueno tau function and extend it on the nonisomonodromic times. Using the underlying Hamiltonian description of the problem we rewrite this extension in terms of the action type integral. Such action integral representation is suitable for evaluation of constant term in the asymptotics. |