## schedule

**Courses (In english)**

**(Universidad de Buenos Aires)**

*Microscopic approximation of the Burgers equation,*Pablo Ferrari**Abstract:**In this course I will give an elementary proof of the convergence of the rescaled TASEP to the Burgers equation in the shock and rarefaction fan cases. One of the main tools is the use of TASEP second class particles as microscopic characteristics (as in the Burgers equation). I will also discuss the multispeed TASEP process. TASEP: totally asymmetric simple exclusion process. Burgers equation: u_t = -(u(1-u))_x.

**(MIT)**

*Topological expansions,*Alice Guionnet**Abstract:**Topological expansions were first exhbit by t'Hooft and Brezin-Itzykson-Parisi-Zuber for matrix intgerals. It turns out that they can also be derived for many large dimension interacting system, allowing the study of the underlying point process, such as the famous beta-models. During this course we will describe the theory of topologial expansions and their extensions, as well as applications to point processes.

**Real zeros of random polynomials****(Indian Institute of Science)**

*,*Manjunath Krishnapur**(Universitat de Barcelona)**

*Fekete points, an overview,*Joaquim Ortega Cerdà**Abstract:**I will present an overview of the distribution of Fekete points in several contexts. These are points that minimize a certain energy and that appear naturally in problems ranging from electrostatics to approximation theory. We will overview the results in different contexts, from the clasical weighted Fekete points in the plane to more elaborated versions in complex and Riemannian manifolds. We will explore how the conection to determinantal point processes can be enlightening in their study.

**(UPEC)**

*Two-scale Gamma convergence for Coulomb gases and weighted Fekete sets,*Etienne Sandier**Abstract:**In this mini-course we will firstly present recent results on the local behaviour of weighted Fekete sets and on the local law of Coulomb gases. These rely on the computation of the second term in the expansion by Gamma-convergence of the interaction energy of the points. This second term is the energy of the physical system known as a jellium, and we will present some of its properties and if time allows, results obtained in collaboration with Y.Ge on this topic.

**(Institute for Low Temperature Physics)**

*Local regimes for beta matrix models,*Mariya Shcherbina

*Distribution of zeroes of entire functions represented by pseudo-random and random Taylor Series***(Tel Aviv University)**

*,*Mikhail Sodin**Abstract:**The asymptotic behaviour and distribution of zeroes of various classes of Taylor series with random and pseudo-random coefficients is governed by certain autocorrelations in relatively short windows. Using this guiding principle, we discuss various examples of pseudo-random and random sequences and, in particular, answer some questions posed by Chen and Littlewood in 1967. We proceed with a somewhat related result that tells that if a spectral measure of a stationary random {0, 1}-valued sequence has a gap in its support,

then the sequence is periodic. The lectures will be mainly based on a joint work with Alexander Borichev and

Alon Nishry posted in arXiv (September 2014).

**(University of Toronto)**

*The spectra of random sparse graphs,*Balint Virag**Abstract:**This short introduction will cover some old and new results. Why is the spectrum of a graph interesting? When does it contain atoms? How continuous is the spectral measure? When are eigenvectors localized / delocalized?

**(Weizmann Institute of Science)**

*Extremal processes,*Ofer Zeitouni**Abstract:**We discuss the structure of extremal processes in nodels exhibiting long-range correlations, emphasizing the role of decorated Poisson processes.

**Invited talks (In english)**

**Random eigenvalues, OCP’s , and Ward identities,****Yacin Ameur**(Lund University)

**Abstract:**Ward identities are exact identities satisfied by the intensity functions of a particle system. They can be derived using reparametrization invariance of the partition function. Whereas Ward identities have been well-known in physical field theories, they were, to the best of my knowledge, overlooked by mathematicians until Johansson applied them in a one-dimensional situation, to study fluctuations of linear statistics of one-dimensional Coulomb gases. The corresponding two-dimensional theory is currently an active area of research. I will overview some of the developments.

**(UDELAR)**

*Distributing points on the sphere and complexity,*Diego Armentano**Abstract:**The aim of this talk is to show how interesting problems of minimal energy configurations over the sphere arise from the study of the complexity of finding a root of a system of polynomials.

**(Universidad de Buenos Aires)**

*Metastability in 1d Zero Range Processes*, Inés Armendáriz**Abstract:**Zero-range processes with decreasing jump rates are known to exhibit condensation, where a finite fraction of all particles concentrates on a single lattice site when the total density exceeds a critical value. We study such a process on a one-dimensional lattice in the thermodynamic limit with fixed, super-critical particle density. We show that the process exhibits metastability with respect to the condensate location, i.e. the suitably accelerated process of the rescaled condensate location converges to a limiting Markov process on the unit torus. This process has stationary, independent increments and the rates are characterized by the scaling limit of capacities of a single random walker on the lattice. Joint work with S. Grosskinsky and M. Loulakis

**(King's College London)**

*A hyperbolic hole theorem,*Jerry Buckley**Abstract:**The hyperbolic Gaussian analytic function (GAF) is a random holomorphic function on the unit disc. It satisfies the remarkable property that the distribution of its zero set is invariant under automorphisms of the discs, and it is essentially the only GAF with this property. A hole is the event that there are no zeroes in a given hyperbolic disc. I will discuss the asymptotic decay of the probability of this event, under various regimes. Joint work with A. Nishry, R. Peled and M. Sodin.

*Large deviations for the empirical field of Coulomb and Riesz systems,*Thomas**Leblé**(Université Pierre et Marie Curie Paris VI)

**Abstract:**We study a system of N particles with Coulomb/Riesz pairwise interactions under a confining potential. After rescaling we deal with a microscopic quantity, the associated empirical point process, for which we give a large deviation principle whose rate function is the sum of a relative entropy and of the

*renormalized energy*defined by Sandier-Serfaty. We also present applications to point processes emerging from random matrix theory. This is joint work with S.Serfaty.

*CLT for random trigonometric polynomials*,**José León**(Universidad Central de Venezuela, INRIA de Grenoble)

**Abstract:**Let us consider two sequences {a_n} and {b_n} of iid N(0,1) random variables. A problem with certain history, consists of the study of the asymptotic behavior of the roots of the stationary random trigonometric polynomials

X_N(t)=\frac{1}{\sqrt{N}} \sum_{n=1}^N a_n sin nt + b_n cos nt,

and also the same study for the non stationary ones

Y_N(t)=\frac{1}{\sqrt N} \sum_{n=1}^N a_n cos nt.

In the first part of this talk we will show that if X is the mean zero stationary Gaussian process with covariance function r_X(t)=\frac{\sin t}{t}. The number of roots of a rescaled modification of X_N can be approximated in the Itô-Wiener Chaos by the number of roots on large intervals of X. Thus the CLT for the crossings of zero of X allows the same theorem for the roots of X_N. In the second part we will consider the non stationary case of Y_N. We also obtain, in a more involved manner, a CLT for the roots of such a process. The process X also appears in the proof. This is a joint work with J-M Azaïs and F. Dalmao.

*Tacnode kernels and Lax systems for the Painlevé II equation,***Liechty**(DePaul University)

**Abstract:**The tacnode process is a determinantal process which was first studied by three separate groups of authors around 2010. One of those groups, Delvaux, Kuijlaars, and Zhang, described the kernel defining the process in terms of a new Lax system for the Painlevé II equation (PII) of size 4x4. I will discuss how this Lax system is related to the 2x2 Lax pair for PII studied by Flaschka and Newell in 1980. As a result, we find new formulas for various tacnode kernels and related kernels in random matrix theory. This is joint work with Dong Wang.

*Hole probability and conditional distribution of random zeros and eigenvalues,*Alon**Nishry**(University of Michigan)

**Abstract:**A 'Hole' is an event where a point process has no points inside a given domain. We are interested in the asymptotics of the probability of the hole event for large domains. In addition, we try to describe the conditional distribution of the point process, given the hole event. I will consider two particular cases, (the zeros of) the 'flat' Gaussian entire function and the Ginibre ensemble.

*Non-intersecting Brownian bridges and the Laguerre Orthogonal Ensemble*,**Remenik**(Universidad de Chile)

**Abstract:**The study of ensembles of non-intersecting paths has attracted a lot of interest in the last decade, in large part due to their connection with the KPZ universality class and random matrix theory (RMT). In many cases, the interest lies in studying the asymptotic fluctuations of the system, as the number of paths N goes to infinity, and showing that their size and distribution coincide with those appearing in RMT. In this talk I will focus instead in the case of finite N, in the context of studying a system of N non-intersecting Brownian bridges (starting and ending at the origin). I will present a result which shows that the squared maximal height of the top path in this system is distributed as the top eigenvalue of a random matrix drawn from the Laguerre Orthogonal Ensemble. This result can be thought of as a discrete version of K. Johansson’s result that the supremum of the Airy2 process minus a parabola has the Tracy-Widom GOE distribution. The result can also be recast in terms of the probability that the top curve of the stationary Dyson Brownian motion hits an hyperbolic cosine barrier. This is joint work with Gia Bao Nguyen.

*Topologies of nodal sets of random band-limited functions***, Igor Wigman**(King's College London)

**Abstract:**This work is joint with Peter Sarnak. It is shown that the topologies and nestings of the zero and nodal sets of random (Gaussian) band limited functions have universal laws of distribution. Qualitative features of the supports of these distributions are determined. In particular the results apply to random monochromatic waves and to random real algebraic hyper-surfaces in projective space.

**Short talks (In english)**

**Fractional Poisson process: Weak convergence and applications,**

**Hector Araya**(PUCV - UTFSM - UV)

**Abstract:**In this paper we study Donsker Type Theorem for fractional Poisson Process (fPp). We present the random walk discretization and the convergence theorem related to the Skorohod Topology. This allows to simulate trajectories and to apply in a long memory ARCH model. This is a joint work with Natalia Bahamonde, Soledad Torres, and Frederi Viens.

**(University of Western Center Lisandro Alvarado, Venezuela)**

*, Nestor Escudero*

Spectral Analysis of Waves SeaSpectral Analysis of Waves Sea

Abstract: Spectral analysis is the technical process of complex signal decomposition into simpler parts. Many physical processes are better described as a sum of individual frequency individual components, alternatively a signal can be divided into short segments and spectrum analysis can be applied to these individual segments. From the point of view of the study of Waves Sea, the spectrum plays a vital role and is interpreted as the energy in the time series during the period. This study is based on analysis of the spectrum associated data in the North Sea in 1999, in the North Alwyn platform stored in the Heriot-Watt University in Edinburgh, with separate periods of 20 minutes, the amount of periods was 244, for this study, the percentage that represent noise was the 3 % of energy and not taken for the study. The spectrum was divided into 12 subintervals and autoregressive models were fitted under the scheme Box-Jenskin and Reinsel, the model was studied until the best possible approximation for each subinterval by the statistical properties obtained by each model, together with the respective forecasts. Then, as an important part in the study, the vector autoregressive models (VAR) were built to explain in more specific the behavior of the storm, finding VAR models with each pair of adjoining intervals, its stability, good fit and the corresponding forecasts were proved.

**(University of Buenos Aires)**

*Metastability for small random perturbations of a PDE with blow-up,*Santiago Saglietti**Abstract:**We consider the stochastic PDE

u_t = u_{xx} + u^p + \epsilon \dot{W}

with homogenous Dirichlet boundary conditions, where p > 1, \epsilon > 0 is a small fixed parameter and \dot{W} stands for space-time white noise. It is well known that the associated deterministic PDE (i.e. \epsilon = 0 in the equation above) admits exactly one asymptotically stable equilibrium and a countable family of unstable equilibria with increasing energy. Furthermore, for certain initial conditions it can be shown that the solution of the deterministic PDE explodes in finite time. We show that, for p < 5 and initial conditions in the domain of attraction of the asymptotically stable equilibrium, the solution X_\epsilon of the SPDE satisfies in the limit as \epsilon tends to zero the classical description of metastability featured in [1]: the averages of X_\epsilon remain stable and close to the equilibrium up until the explosion time which, when suitably rescaled, converges in distribution to an exponential random variable. Furthermore, for certain initial conditions in the domain of explosion (and any value of p > 1) we show the continuity of the explosion time as \epsilon tends to zero.

[1] Galves, Antonio; Olivieri, Enzo; Vares, Maria Eulália. Metastability for a class of dynamical systems subject to small random perturbations. Ann. Probab. 15 (1987), no. 4, 1288-1305.

**(Mathematics Insitute, University of Warwick)**

*Orthogonal polynomials vs. cyclic functions*, Daniel Seco**Abstract:**A function f is cyclic (in a space of analytic functions X) if the polynomial multiples of f form a dense subspace of X. I will present a recent work with Bènèteau, Liaw and Sola, where we describe cyclicity as a question about orthogonal polynomials in some associated spaces. We also study connections with reproducing kernels, and between the zeros of several families of polynomials.

*Long-range order in random 3-colorings of Zd,***Yinon Spinka**(Tel Aviv University)

**Abstract:**Consider a random coloring of a bounded domain in Zd with the probability of each coloring F proportional to exp(-β*N(F)), where β>0 is a parameter (representing the inverse temperature) and N(F) is the number of nearest neighboring pairs colored by the same color. This is the anti-ferromagnetic 3-state Potts model of statistical physics, used to describe magnetic interactions in a spin system. The Kotecký conjecture is that in such a model, for d≥3 and high enough β, a sampledcoloring will typically exhibit long-range order, placing the same color at most of either the even or odd vertices of the domain. We give the first rigorous proof of this fact for large d. This extends previous works of Peled and of Galvin, Kahn, Randall and Sorkin, who treated the case β equals infinity.

The main ingredient in our proof is a new structure theorem for 3-colorings which characterizes the ways in which different "phases" may interact, putting special emphasis on the role of edges connecting vertices of the same color. We also discuss several related conjectures. No background in statistical physics will be assumed and all terms will be explained thoroughly. Joint work with Ohad Feldheim.