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Monday, 14 January 2019
Time Speaker Title Resources
09:45 to 10:40 Arup Bose Smallest singular value and limit eigenvalue distribution of a class of non-Hermitian random matrices with statistical application

Suppose $X$ is an $N \times n$ complex matrix whose entries are centered, independent, and identically distributed random variables with variance $1/n$ and whose fourth moment is of order ${\mathcal O}(n^{-2})$. We first consider the non-Hermitian matrix $X A X^* - z$, where $A$ is a deterministic matrix whose smallest and largest singular values are bounded below and above respectively, and $z\neq 0$ is a complex number. A symptotic probability bounds for the smallest singular value of this model are obtained in the large dimensional regime where $N$ and $n$ diverge to infinity at the same rate.

We then consider the special case where $A = J = [1_{i-j = 1\mod n} ]$ is a circulant matrix. Using the result of the first part, it is shown that the limit eigenvalue distribution of $X J X^*$ exists in the large dimensional regime, and we determine this limit explicitly. A statistical application of this result devoted towards testing the presence of correlations within a multivariate time series is considered. Assuming that $X$ represents a $CC^N$-valued time series which is observed over a time window of length $n$, the matrix $X J X^*$ represents the one-step sample autocovariance matrix of this time series. Guided by the result on the limit spectral measure of this matrix, a whiteness test against an MA correlation model on the time series is introduced. Numerical simulations show the excellent performance of this test.

This is joint work with Walid Hachem.

10:40 to 11:00 -- Tea/coffee break
11:00 to 11:55 Anirban Basak Sharp transition of invertibility of sparse random matrice

Consider an $n \times n$ matrix with i.i.d.~Bernoulli($p$) entries. It is well known that for $p= \Omega(1)$, i.e.~$p$ is bounded below by some positive constant, the matrix is invertible with high probability. If $p \ll \frac{\log n}{n}$ then the matrix contains zero rows and columns with high probability and hence it is singular with high probability. In this talk, we will discuss the sharp transition of the invertibility of this matrix at $p =\frac{\log n}{n}$. This phenomenon extends to the adjacency matrices of directed and undirected Erd\H{o}s-R\'{e}nyi graphs, and random bipartite graph. This is joint work with Mark Rudelson.

12:00 to 12:55 David Renfrew Eigenvalues of random n on-Hermitian matrices and randomly coupled differential equations

We consider large random matrices with centered, independent entries but possibly different variances and compute the limiting distribution of eigenvalues. We then consider applications to long time asymptotics for systems of critically coupled differential equations with random coefficients.

12:55 to 14:00 -- Lunch
14:00 to 15:00 Charles Bordenave High trace methods in random matrix theory (Remote Talk)

In 1955, Eugene Wigner has established the semi-circular law by computing expected traces of random matrices. In 1981, Füredi and Komlos have refined the computation of Wigner and studied the spectral radius of random matrices. Since then, there have been numerous successful extensions of their approach notably in connexion with the non-backtracking matrices. In this mini-course, we will introduce the high trace method of Furedi-Komlos and present some its latest developments: the use of non-backtracking matrices, the tangle-free random graphs and the comparison argument of Bandeira and Van Handel.

15:00 to 15:30 -- Tea/ Coffee break
16:00 to 17:00 Sourav Chatterjee Yang-Mills for mathematicians (Lecture 1) - Ramanujan Lectures

Making sense of quantum field theories is one of the most important open problems of modern mathematics. It is not very well known in the mathematics community that many small parts of this big problem are in fact well-posed questions in probability theory. In this talk I will describe a number of probabilistic open questions, which, if solved, would contribute greatly towards the goal of rigorous construction of quantum field theories. Specifically, I will discuss Yang-Mills theories, lattice gauge theories, quark confinement, mass gap and gauge-string duality, all as problems in probability

Tuesday, 15 January 2019
Time Speaker Title Resources
10:30 to 11:25 Philip Wood Outliers in the spectrum for products of independent random matrices

For fixed positive integers m, we consider the product of m independent n by n random matrices with iid entries as in the limit as n tends to infinity. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the m-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Terence Tao for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations.

11:30 to 12:25 Arijit Chakrabarty Spectra of Adjacency and Laplacian Matrices of inhomogeneous Erdős–Rényi Graphs

Inhomogeneous Erd\H{o}s-R\'enyi random graphs $\mathbb{G}_N$ on $N$ vertices in the non-dense regime are studied. The edge between the pair of vertices $\{i,j\}$ is retained with probability $\varepsilon_N f(\tfrac{i}{N},\tfrac{j}{N})$, $1 \leq i,j \leq N$, independently of other edges, where $f\colon\,[0,1]\times [0,1] \to [0,\infty)$ is a continuous function such that $f(x,y)=f(y,x)$ for all $x,y \in [0,1]$. We study the empirical distribution of both the adjacency matrix $A_N$ and the Laplacian matrix $\Delta_N$ associated with $\mathbb{G}_N$ in the limit as $N \to \infty$ when $\lim_{N\to\infty} \varepsilon_N = 0$ and $\lim_{N\to\infty} N\varepsilon_N = \infty$. In particular, we show that the empirical distributions of $(N\varepsilon_N)^{-1/2} A_N$ and $(N\varepsilon_N)^{-1/2} \Delta_N$ converge to deterministic limits weakly in probability. For the special case where $f(x,y) = r(x)r(y)$ with $r\colon\, [0,1] \to [0,\infty)$ a continuous function, we give an explicit characterisation of the limiting distributions.

12:55 to 14:00 -- Lunch
14:00 to 15:00 Charles Bordenave High trace methods in random matrix theory (Remote Talk) - Lecture 2

In 1955, Eugene Wigner has established the semi-circular law by computing expected traces of random matrices. In 1981, Füredi and Komlos have refined the computation of Wigner and studied the spectral radius of random matrices. Since then, there have been numerous successful extensions of their approach notably in connexion with the non-backtracking matrices. In this mini-course, we will introduce the high trace method of Furedi-Komlos and present some its latest developments: the use of non-backtracking matrices, the tangle-free random graphs and the comparison argument of Bandeira and Van Handel.

Wednesday, 16 January 2019
Time Speaker Title Resources
09:45 to 10:40 Fanny Augeri Nonlinear large deviations bounds with applications to sparse Erdös-Renyi graphs

In this talk, I will present the framework of the so-called nonlinear large deviations introduced by Chatterjee and Dembo. In a seminal paper, they provided a sufficient criterion in order that the large deviations of a function on the discrete hypercube to be due by only changing the mean of the background measure. This sufficient condition was formulated in terms of the complexity of the gradient of the function of interest. I will present general nonlinear large deviation estimates similar to Chatterjee-Dembo's original bounds except that we do not require any second order smoothness. The approach relies on convex analysis arguments and is valid for a broad class of distributions. Then, I will detail an application of this nonlinear large deviations bounds to the problem of estimating the upper tail of cycles counts in sparse Erdös-Renyi graphs down to the connectivity parameter $n^{-1/2}$.

10:40 to 11:00 -- Tea/coffee break
11:00 to 11:55 Nanda Kishore Reddy Eigenvalues of product random matrices

Products of random matrices have always been a topic of interest in  Mathematics, Physics and Statistics for various reasons. In this talk, we shall discuss, along with their relevance, the  exact eigenvalue distributions of certain product random matrix models and also the  asymptotic behaviour  of the eigenvalues of products of random matrices with the  matrix sizes fixed  and the number of matrices in the product increasing.

12:00 to 12:55 Zhou Fan Tracy-Widom at each edge of real covariance and MANOVA estimators

We study the sample covariance matrix for real-valued data with general population covariance, as well as MANOVA-type covariance estimators in statistical variance components models under null hypotheses of global sphericity. In the limit as matrix dimensions increase proportionally, the asymptotic spectra of such estimators may have multiple disjoint intervals of support, possibly intersecting the negative half line. We show that the distribution of the extremal eigenvalue at each regular edge of the support has a GOE Tracy-Widom limit. Our proof extends a universality argument of Lee and Schnelli, replacing a continuous Green function flow by a discrete Lindeberg swapping scheme. This is joint work with Iain M. Johnstone.

12:55 to 14:00 -- Lunch
14:00 to 15:00 Charles Bordenave High trace methods in random matrix theory (Remote Talk) - Lecture 3
15:00 to 15:30 -- Tea/ Coffee break
16:00 to 17:00 Sourav Chatterjee Gauge-string duality in lattice gauge theories (Ramanujan Lectures) - Lecture 2

Quantum gauge theories are the mathematically ill-defined building blocks of the Standard Model of quantum mechanics. String theories, on the other hand, were built to serve as models of quantum gravity. Physicists have long been aware of the existence of a duality between quantum Yang-Mills theories and string theories. This is sometimes called “gauge-string duality” or “gauge-gravity duality”. Making sense of the duality formulas is still well beyond the reach of rigorous mathematics, partly because the models in question have not yet been rigorously defined. In this talk I will present a rigorously proved version of gauge-string duality in a discrete setting. Specifically, I will take a lattice gauge theory, which is a discrete approximation of a quantum gauge theory, and prove an explicit duality with a kind of string theory on the lattice. The duality has the appearance of a natural discrete analog of the formulas conjectured for the continuum models. The question of proving a continuum version of this duality remains open. These and other open questions will be discussed. (Partly based on joint work with Jafar Jafarov.)

17:00 -- Banquet
Thursday, 17 January 2019
Time Speaker Title Resources
10:30 to 11:25 Ofer Zeitouni On eigenvectors of non-normal random matrices

What is the typical inner product between eigenvectors of non-normal matrices from the invariant ensembles with density proportional to $e^{-\mbox{ \rm Tr} V(XX^*)} dX$? In the Ginibre case (i.e. $V(x)=x$), when the eigenvectors are chosen to correspond to specific eigenvalues,
a CLT can be proved (after proper re-scaling). In the general case, the scale is known, but no limit law is known. I will describe the known (to me)  results and their proofs. I will also  describe a particular large deviations problem that seems essential for th e general case.
(Joint work with Florent Benaych-Georges)

11:30 to 12:25 Subhroshekhar Ghosh Two manifestations of rigidity in point sets: forbidden regions and maximal degeneracy

A point process is said to be "rigid" if its local observables are completely determined (as deterministic functions of) the point configuration outside a local neighbourhood. For example, it has been shown in earlier work that, in the Ginibre ensemble (a.k.a. the 2D Cou lomb gas at inverse temperature beta=2), the point configuration outside any bounded domain determines, almost surely, the number of points in the domain.
In this talk, we will explore two recent manifestations of such rigidity phenomena. For the zeros of the planar Gaussian analytic function, we show that outside every large "hole", there is a "forbidden region" which contains a vanishing density of points. This should be seen in contrast with the corresponding situation for classically understood models (e.g. random matrix ensembles), where no such effects are known to occur.
In the second part of the talk, we will consider "stealthy" hyperuniform systems, which are systems whose structure function (i.e., the Fourier transform of the two-point correlation) vanishes near the origin. We show that such systems exhibit "maximal degeneracy", namely the points outside a bounded domain determine, almost surely, the entire point con figuration inside the domain. En route, we establish a conjecture of Zhang, Stillinger and Torquato that such systems have (deterministically) bounded holes.
Based on joint works with Joel Lebowitz and Alon Nishry.

12:55 to 14:00 -- Lunch
14:00 to 15:00 Charles Bordenave High trace methods in random matrix theory (Remote Talk) - Lecture 4
Friday, 18 January 2019
Time Speaker Title Resources
09:45 to 10:40 Anish Mallick Regularity properties of LSD for certain families of random patterned matrices

In random matrix theory, after defining a family of random matrix one of the first question one asks is about the existence and regularity of limiting empirical spectral distrib ution (LSD). Here, I will talk about the absolute continuity and bound on the density of LSD for random Hankel and Toeplitz matrices.

10:40 to 11:00 -- Tea/coffee break
11:00 to 11:55 Satya Majumdar Rotating trapped fermions in 2d and the complex Ginibre ensemble

We establish an exact mapping between the positions of N non interacting fermions in a 2d rotating harmonic trap in its ground-state and the eigenvalues of the NxN complex Ginibre ensemble of Random Matrix Theory (RMT). Using RMT techniques, we make precise predictions for the statistics of the positions of the fermions, both in the bulk as well as at the edge of the trapped Fermi gas. In addition, we compute exactly, for any finite N, the R\'enyi entanglement entropy and the number variance of a disk of radius r in the ground-state. We show that while these two quantities are proportional to each other in the (extended) bulk, this is no longer the case very close to the trap center nor at the edge. Near the edge, and for large N, we provide exact expressions for the scaling functions associate d with these two observables.

12:00 to 12:55 Shirshendu Ganguly Polymer geometry in the large deviation regime via eigenvalue rigidity

Polymer weights in certain two dimensional exactly solvable models of last passage percolation in the KPZ universality class are known to exhibit remarkable dis- tributional equalities with eigenvalues of well known random matrix ensembles and other determinantal processes. A general goal of the talk will be to explore consequences of recent advances in the study of rigidity properties such point processes in the context of polymer geometry. We will discuss results about precise transversal uctuation behavior of the polymer in upper and lower tail large deviation regimes using various random matrix theory inputs as well as geometric arguments, sharpening a result obtained by Deuschel and Zeitouni (1999) and addressing an open question left by them. Time permitting, we shall also discuss how some of these results extend beyond the exactly solvable settings.

12:55 to 14:00 -- Lunch
14:00 to 15:00 Sourav Chatterjee Constructing a solution of the 2D Kardar-Parisi-Zhang equation (Ramanujan Lectures) - Lecture 3

The Kardar-Parisi-Zhang (KPZ) equation has become accepted as the canonical model for the growth of random surfaces. While the 1D KPZ equation has now a vast amount of rigorous mathematical work behind it, the physically important case of 2D surfaces has remained mathematically intractable. I will describe a first step towards constructing a solution of the 2D KPZ equation, by showing the existence of certain subsequential scaling limits if the parameters of the equation are renormalized in a suitable way. Many open questions remain, and these will be discussed. (Based on recent joint work with Alex Dunlap.)

15:00 to 15:30 -- Tea/ Coffee break
18:00 -- Cultural Program
Monday, 21 January 2019
Time Speaker Title Resources
09:45 to 10:45 Tomohiro Sasamoto Integrable stochastic interacting systems (Lecture - 1)

Stochastic interacting particle systems show many intriguing phenomena due to the interaction among particles and have wide applications in various fields of science, but in general it is quite difficult to study their properties in detail. Over the last few decades, however, it has been gradually recognized that certain stochastic interacting particle systems can be “solved exactly”, meaning that they admit explicit calculations of various probabilities and expectation values, and behind this tractability lies the integrability of these systems. In particular there have been remarkable progress in the understanding of growth and transport models in the Kardar-Parisi-Zhang (KPZ) universality class, which have turned out to have deep connections with random matrix theory, representation theory, special functions and so on.

10:45 to 11:00 -- Discussions
11:00 to 11:30 -- Tea/coffee break
11:30 to 12:30 Mustazee Rahman On shocks in TASEP

The TASEP particle system runs into traffic jams when the initial particle density increases in the direction of flow. It serves as a microscopic model of shocks in Burgers' equation. I will describe work with Jeremy Quastel on a specialization of the TASEP shock, where we identify the microscopic shock process by using determinantal formulae for the correlation functions of particles. The process is described in terms of the Tracy-Widom laws, and, interestingly, our proof relies on connections to stochastic integrable systems that arise within KPZ models.

12:55 to 14:00 -- Lunch
14:00 to 15:00 Sunil Chhita A (2+1)-dimensional Anisotropic KPZ growth model with a smooth phase

Stochastic growth processes in dimension (2+1) were conjectured by D. Wolf, on the basis of renormalization-group arguments, to fall into two distinct universality classes known as the isotropic KPZ class and the anisotropic KPZ class (AKPZ). The former is ch aracterized by strictly positive growth and roughness exponents, while in the AKPZ class, fluctuations are logarithmic in time and space. These classes are determined by the sign of the determinant of the Hessian of the speed of growth.

It is natural to ask (a) if one can exhibit interesting growth models with "smooth" stationary states, i.e., with O(1) fluctuations (instead of logarithmically or power-like growing, as in Wolf's picture) and (b) what new phenomena arise when the speed of growth is not smooth, so that its Hessian is not defined. These two questions are actually related and in this talk, we provide an answer to both, in a specific framework. This is joint work with Fabio Toninelli (CNRS and Lyon 1).

15:00 to 15:30 -- Tea/coffee break
15:30 to 16:30 Herbert Spohn Generalized Hydrodynamics and the classical Toda chain

In the context of integrable quantum many-body systems, much progress has been achieved in deriving and analysing the infinite set of coupled local conservation laws constituting "generalized hydrodynamics". In my presentation I will outline the scheme for the classical Toda chain exploring unexpected connections to random matrix theory.

Tuesday, 22 January 2019
Time Speaker Title Resources
09:45 to 10:45 Tomohiro Sasamoto Integrable stochastic interacting systems (Lecture - 2)

Stochastic interacting particle systems show many intriguing phenomena due to the interaction among particles and have wide applications in various fields of science, but in general it is quite difficult to study their properties in detail. Over the last few decades, however, it has been gradually recognized that certain stochastic interacting particle systems can be “solved exactly”, meaning that they admit explicit calculations of various probabilities and expectation values, and behind this tractability lies the integrability of these systems. In particular there have been remarkable progress in the understanding of growth and transport models in the Kardar-Parisi-Zhang (KPZ) universality class, which have turned out to have deep connections with random matrix theory, representation theory, special functions and so on.

10:45 to 11:00 -- Discussions
11:00 to 11:30 -- Tea/coffee break
11:30 to 12:30 Sakuntala Chatterjee Dynamics of coupled modes for sliding particles on a fluctuating landscape

We study a system of hardcore particles sliding on a fluctuating potential energy landscape, whose dynamics follows KPZ equation. While the particles slide down along the local slope of the landscape and tend to settle in the local valleys, they also modify the local dynamics of the landscape around their positions. By tuning the coupling parameters between the landscape and the particles, we derive a phase diagram for the system which consists of various kinds of nonequilibrium ordered and disordered phases. In this talk, I will mainly focus on the dynamics in the disordered phase. The coupled dynamics of the two conserved fields, {\slviz.} the density of the sliding particles, and the local tilt or height gradient of the landscape, gives rise to various different universality classes in the system and we aim to use the formalism of non-linear fluctuating hydrodynamics to unravel them. However, even in the disordered phase the system does not satisfy product measure and shows short-ranged correlations. The exact steady state measure is not known and we do not have exact expressions for the particle and tilt currents. Therefore, we use approximate expressions for the currents within mean field theory and check our predictions against numerical simulations. Amo ng the non-KPZ universality classes, we observe 5/3 Levy, diffusive and modified KPZ for different parameter values. Although our analytics predict golden mean and 3/2 Levy as well, we are not able to observe them in our simulations because of strong finite size effects.

12:55 to 14:00 -- Lunch
Wednesday, 23 January 2019
Time Speaker Title Resources
09:45 to 10:45 Tomohiro Sasamoto Integrable stochastic interacting systems (Lecture - 3)

Stochastic interacting particle systems show many intriguing phenomena due to the interaction among particles and have wide applications in various fields of science, but in general it is quite difficult to study their properties in detail. Over the last few decades, however, it has been gradually recognized that certain stochastic interacting particle systems can be “solved exactly”, meaning that they admit explicit calculations of various probabilities and expectation values, and behind this tractability lies the integrability of these systems. In particular there have been remarkable progress in the understanding of growth and transport models in the Kardar-Parisi-Zhang (KPZ) universality class, which have turned out to have deep connections with random matrix theory, representation theory, special functions and so on.

10:45 to 11:00 -- Discussions
11:00 to 11:30 -- Tea/coffee break
11:30 to 12:30 Shirshendu Ganguly Fractal Properties of Coupled Polymer Weight Profiles via Coalescence of Geodesics
12:55 to 14:00 -- Lunch
14:00 to 15:00 Nikolaos Zygouras The two-dimensional KPZ and other marginally relevant disordered systems

In joint works with Francesco Caravenna and Rongfeng Sun we have initiated a program of studying scaling limits of disordered systems, where disorder has a “marginally relevant” effect. In the language of stochastic analysis and renormalisa- tion theory this corresponds to studying randomly perturbed models at the “critical dimension”. One such model is the two-dimensional KPZ equation. A consequence of our previous works has been that the two-dimensional KPZ with the noise mollified in space on scale ∊ and scaled as βˆ√ p | log | undergoes a phase transition with an explicit critical point βˆ c = √ 2π. In a more recent work we show that the so- lution to the mollified and renormalised equation has a unique limit in the entire subcritical regime βˆ ∈ (0, βˆ c), which we have identified as the solution to an additive Stochastic Heat Equation, establishing so-called Edwards-Wilkinson fluctuations.

15:00 to 15:30 -- Tea/coffee break
15:30 to 17:00 -- Short Talks
Thursday, 24 January 2019
Time Speaker Title Resources
09:45 to 10:45 Tomohiro Sasamoto Integrable stochastic interacting systems (Lecture - 4)

Stochastic interacting particle systems show many intriguing phenomena due to the interaction among particles and have wide applications in various fields of science, but in general it is quite difficult to study their properties in detail. Over the last few decades, however, it has been gradually recognized that certain stochastic interacting particle systems can be “solved exactly”, meaning that they admit explicit calculations of various probabilities and expectation values, and behind this tractability lies the integrability of these systems. In particular there have been remarkable progress in the understanding of growth and transport models in the Kardar-Parisi-Zhang (KPZ) universality class, which have turned out to have deep connections with random matrix theory, representation theory, special functions and so on.

10:45 to 11:00 -- Tea/coffee break
11:00 to 11:55 Sanjay Ramassamy Dimers and circle patterns

The dimer model is a model from statistical mechanics corresponding to random perfect matchings on graphs. Circle patterns are a class of embeddings of planar graphs such that every face admits a circumcircle. In this talk I describe a correspondence between dimer models on planar bipartite graphs and circle pattern embeddings of these graphs. As special cases of this correspondence we recover the Tutte embeddings (a.k.a harmonic embeddings) for resistor networks and the s-embeddings for Ising models. This correspondence is also the key for studying Miquel dynamics, a discrete integrable system on circle patterns. This is joint work with Richard Kenyon (Brown University), Wai Yeung Lam (Brown University) and Marianna Russkikh (University of Geneva).

12:00 to 12:55 Eunghyun Lee Some exact formulas in the integrable particle models with multi-species

In this talk, we treat the extensions to the multi-species version of some particles models in the integrable probability. We will see how to check the integrabilities of the models and how to find the transition probabilities. In particular, we extend Chatterjee and Schutz's result (2010, JSP) on the TASEP with second class particles which gives some determinantal formulas to the ASEP with multi-species.

12:55 to 14:00 -- Lunch
Friday, 25 January 2019
Time Speaker Title Resources
09:45 to 10:45 Tomohiro Sasamoto Integrable stochastic interacting systems (Lecture - 5)

Stochastic interacting particle systems show many intriguing phenomena due to the interaction among particles and have wide applications in various fields of science, but in general it is quite difficult to study their properties in detail. Over the last few decades, however, it has been gradually recognized that certain stochastic interacting particle systems can be “solved exactly”, meaning that they admit explicit calculations of various probabilities and expectation values, and behind this tractability lies the integrability of these systems. In particular there have been remarkable progress in the understanding of growth and transport models in the Kardar-Parisi-Zhang (KPZ) universality class, which have turned out to have deep connections with random matrix theory, representation theory, special functions and so on.

10:45 to 11:00 -- Tea/coffee break
11:00 to 12:55 -- Short Talks
12:55 to 14:00 -- Lunch
Monday, 28 January 2019
Time Speaker Title Resources
10:00 to 11:00 Daniel Remenik The KPZ fixed point - (Lecture 1)

The Kardar-Parisi-Zhang (KPZ) universality class is a broad class of models coming from mathematical physics which includes random interface growth, directed random polymers, interacting particle systems, and random stirred fluids. These models share a very special and rich asymptotic fluctuation behavior, which is loosely characterized by fluctuations which grow like t^{1/3} as time t evolves, decorrelate at a spatial scale of t^{2/3}, and have certain very special limiting distibutions; this fluctuation behavior is model independent but depends on the initial data, and in some important cases it is connected with distributions coming from random matrix theory.

A somewhat vague conjecture in the field was that there should be a universal, scaling invariant limit for all models in the KPZ class, containing all the fluctuation behavior seen in the class. In these lectures I will describe joint work with K. Matetski and J. Quastel [5] where we were able to construct and give a complete description of this limiting process, known as the KPZ fixed point. This limiting universal process is a Markov process, taking values in real valued functions which look locally like Brownian motion.

The construction follows from an novel exact solution for one of the most basic models in the KPZ class, the totally asymmetric exclusion process (TASEP), for arbitrary initial condition. This formula is given as the Fredholm determinant of a kernel involving the transition probabilities of a random walk forced to hit a curve defined by the initial data, and in the KPZ 1:2:3 scaling limit the formula leads in a transparent way to a Fredholm determinant formula given in terms of analogous kernels based on Brownian motion.

Two sets of lecture notes [6,7] serve as a good complement to the mini-course.

References:

1. A. Borodin, P. L. Ferrari, M. Prähofer, and T. Sasamoto. Fluctuation properties of the TASEP with periodic initial configuration. J. Stat. Phys. 129.5-6 (2007), pp. 10551080.
2. A. Borodin, I. Corwin, and D. Remenik. Multiplicative functionals on ensembles of non-intersecting paths. Ann. Inst. H. Poincaré Probab. Statist. 51.1 (2015), pp. 28–58.
3. I. Corwin, J. Quastel, and D. Remenik. Continuum statistics of the Airy 2 process. Comm. Math. Phys. 317.2 (2013), pp. 347–362.
4. J. Quastel, D. Remenik. How flat is flat in random interface growth? To appear in Trans. AMS. arXiv:1606.09228.
5. K. Matetski, J. Quastel and D. Remenik. The KPZ fixed point. arXiv:1701.00018.
6. K. Matetski, J. Quastel. From TASEP to the KPZ fixed point. arXiv:1710.02635.
7. D. Remenik. Course notes on the KPZ fixed point.
8. T. Sasamot o. Spatial correlations of the 1D KPZ surface on a flat substrate. J. Phys. A 38.33 (2005), p. L549.
11:00 to 11:15 -- Discussions
11:15 to 11:45 -- Tea/coffee break
11:45 to 12:45 Leandro Pimentel Ergodicity of the KPZ Fixed Point

The Kardar-Parisi-Zhang (KPZ) fixed point is a Markov process in the space of upper semi-continuous functions, introduced recently by Matetski, Quastel and Remenik (2017). It describes the limit fluctuations of the height function associated to the totally asymmetric simple exclusion process (TASEP), and it is conjectured to be the limit fluctuations of a wide class of 1+1 interface growth process (KPZ universality class). Our main result is that the KPZ fixed point centred at the origin converges in distribution, as time goes to infinity, to a two-sided Brownian motion with zero drift and diffusion coefficient 2. The heart of the proof is the coupling method, that allows us to compare the TASEP height function started from a perturbation of density 1/2 with its invariant counterpart.

12:45 to 14:00 -- Lunch
14:00 to 15:00 Patrik Ferrari Time-time covariance for last passage percolation with generic initial profile

We consider time correlation for KPZ growth in 1+1 dimensions in a neighborhood of a characteristics. We prove convergence of the covariance with droplet, flat and stationary initial profile. In particular, this provides a rigorous proof of the exact formula of the covariance for the stationary case obtained in [SIGMA 12 (2016), 074]. Furthermore, we prove the universality of the first order correction when the two observation times are close and provide a rigorous bound of the error term. This result holds also for random initial profiles which are not necessarily stationary.

15:00 to 15:30 -- Tea/coffee break
15:30 to 16:30 Sunder Sethuraman On Hydrodynamic Limits of Young Diagrams

We consider a family of stochastic models of evolving two-dimensional Young diagrams, given in terms of certain energies, w ith Gibbs invariant measures. Static' scaling limits of the shape functions, under these Gibbs measures, have been shown by several over the years. In this talk, we discuss corresponding dynamical' limits which are less understood. We show that the hydrodynamic scaling limits of the diagram shape functions may be described by different types parabolic PDEs, depending on the energy structure.

Tuesday, 29 January 2019
Time Speaker Title Resources
10:00 to 11:00 Daniel Remenik The KPZ fixed point - (Lecture 2)

The Kardar-Parisi-Zhang (KPZ) universality class is a broad class of models coming from mathematical physics which includes random interface growth, directed random polymers, interacting particle systems, and random stirred fluids. These models share a very special and rich asymptotic fluctuation behavior, which is loosely characterized by fluctuations which grow like t^{1/3} as time t evolves, decorrelate at a spatial scale of t^{2/3}, and have certain very special limiting distibutions; this fluctuation behavior is model independent but depends on the initial data, and in some important cases it is connected with distributions coming from random matrix theory.

A somewhat vague conjecture in the field was that there should be a universal, scaling invariant limit for all models in the KPZ class, containing all the fluctuation behavior seen in the class. In these lectures I will describe joint work with K. Matetski and J. Quastel [5] where we were able to construct and give a complete description of this limiting process, known as the KPZ fixed point. This limiting universal process is a Markov process, taking values in real valued functions which look locally like Brownian motion.

The construction follows from an novel exact solution for one of the most basic models in the KPZ class, the totally asymmetric exclusion process (TASEP), for arbitrary initial condition. This formula is given as the Fredholm determinant of a kernel involving the transition probabilities of a random walk forced to hit a curve defined by the initial data, and in the KPZ 1:2:3 scaling limit the formula leads in a transparent way to a Fredholm determinant formula given in terms of analogous kernels based on Brownian motion.

Two sets of lecture notes [6,7] serve as a good complement to the mini-course.

References:

1. A. Borodin, P. L. Ferrari, M. Prähofer, and T. Sasamoto. Fluctuation properties of the TASEP with periodic initial configuration. J. Stat. Phys. 129.5-6 (2007), pp. 10551080.
2. A. Borodin, I. Corwin, and D. Remenik. Multiplicative functionals on ensembles of non-intersecting paths. Ann. Inst. H. Poincaré Probab. Statist. 51.1 (2015), pp. 28–58.
3. I. Corwin, J. Quastel, and D. Remenik. Continuum statistics of the Airy 2 process. Comm. Math. Phys. 317.2 (2013), pp. 347–362.
4. J. Quastel, D. Remenik. How flat is flat in random interface growth? To appear in Trans. AMS. arXiv:1606.09228.
5. K. Matetski, J. Quastel and D. Remenik. The KPZ fixed point. arXiv:1701.00018.
6. K. Matetski, J. Quastel. From TASEP to the KPZ fixed point. arXiv:1710.02635.
7. D. Remenik. Course notes on the KPZ fixed point.
8. T. Sasamot o. Spatial correlations of the 1D KPZ surface on a flat substrate. J. Phys. A 38.33 (2005), p. L549.
11:00 to 11:15 -- Discussions
11:15 to 11:45 -- Tea/coffee break
11:45 to 12:45 Manas Kulkarni Connections between Classical Calogero-Moser, Log Gas and Random Matrix Theory

We present a deep connection between the classical Calogero-Moser (CM) model, Log-gas (LG) model and Random Matrix Theory (RMT). We show that CM model has some remarkable connections with the 1D LG model. Both models have the same minimum energy configuration with the particle positions given by the zeros of the Hermite potential. Moreover the Hessian describing small oscilla tions around equilibrium are also related for the two models. We explore this connection further by studying finite temperature equilibrium properties of the CM model through Monte-Carlo simulations and comparing them with known LG results. In particular, our findings indicate that the single particle distribution and the marginal distribution of the boundary particle of CM model are also given by Wigner semi-circle and the Tracy-Widom distribution respectively (similar to LG model). Comparisons are made with analytical predictions from the small oscillation theory and we find very good agreement. Parallels are also drawn with rigorous mathematical results from RMT and implications of finite-size as well as finite-temperature effects are observed. We also present some preliminary results on large deviations in CM model by using field theory.

12:45 to 14:00 -- Lunch
14:00 to 15:00 Abhishek Dhar Hydrodynamics and chaos in spin chains: connections to KPZ

The first part of the talk will discuss the predictions of nonlinear fluctuating hydrodynamics for a one-dimensional chain of spins, described by the XXZ Hamiltonian and evolving with Hamiltonian dynamics. One of the interesting features, at low temperatures, is the emergence of "almost" conserved quantities leading to sound modes with KPZ scaling. Numerical results on equilibrium correlation functions, to check the predictions of the theory, will be presented. In the second part we discuss a different quantity which quantifies the chaotic spread and growth of localized perturbations. It is shown that this can be effectively described as a growing one-dimensional interface with KPZ scaling.

15:00 to 15:30 -- Tea/coffee break
Wednesday, 30 January 2019
Time Speaker Title Resources
10:00 to 11:00 Daniel Remenik The KPZ fixed point - (Lecture 3)

The Kardar-Parisi-Zhang (KPZ) universality class is a broad class of models coming from mathematical physics which includes random interface growth, directed random polymers, interacting particle systems, and random stirred fluids. These models share a very special and rich asymptotic fluctuation behavior, which is loosely characterized by fluctuations which grow like t^{1/3} as time t evolves, decorrelate at a spatial scale of t^{2/3}, and have certain very special limiting distibutions; this fluctuation behavior is model independent but depends on the initial data, and in some important cases it is connected with distributions coming from random matrix theory.

A somewhat vague conjecture in the field was that there should be a universal, scaling invariant limit for all models in the KPZ class, containing all the fluctuation behavior seen in the class. In these lectures I will describe joint work with K. Matetski and J. Quastel [5] where we were able to construct and give a complete description of this limiting process, known as the KPZ fixed point. This limiting universal process is a Markov process, taking values in real valued functions which look locally like Brownian motion.

The construction follows from an novel exact solution for one of the most basic models in the KPZ class, the totally asymmetric exclusion process (TASEP), for arbitrary initial condition. This formula is given as the Fredholm determinant of a kernel involving the transition probabilities of a random walk forced to hit a curve defined by the initial data, and in the KPZ 1:2:3 scaling limit the formula leads in a transparent way to a Fredholm determinant formula given in terms of analogous kernels based on Brownian motion.

Two sets of lecture notes [6,7] serve as a good complement to the mini-course.

References:

1. A. Borodin, P. L. Ferrari, M. Prähofer, and T. Sasamoto. Fluctuation properties of the TASEP with periodic initial configuration. J. Stat. Phys. 129.5-6 (2007), pp. 10551080.
2. A. Borodin, I. Corwin, and D. Remenik. Multiplicative functionals on ensembles of non-intersecting paths. Ann. Inst. H. Poincaré Probab. Statist. 51.1 (2015), pp. 28–58.
3. I. Corwin, J. Quastel, and D. Remenik. Continuum statistics of the Airy 2 process. Comm. Math. Phys. 317.2 (2013), pp. 347–362.
4. J. Quastel, D. Remenik. How flat is flat in random interface growth? To appear in Trans. AMS. arXiv:1606.09228.
5. K. Matetski, J. Quastel and D. Remenik. The KPZ fixed point. arXiv:1701.00018.
6. K. Matetski, J. Quastel. From TASEP to the KPZ fixed point. arXiv:1710.02635.
7. D. Remenik. Course notes on the KPZ fixed point.
8. T. Sasamot o. Spatial correlations of the 1D KPZ surface on a flat substrate. J. Phys. A 38.33 (2005), p. L549.
11:00 to 11:15 -- Discussions
11:15 to 11:45 -- Tea/coffee break
11:45 to 12:45 Marc Brachet Crossover of equilibrium correlation time for the 1D Galerkin-truncated Burgers equation in the limit of vanishing noise and dissipation

Classical Galerkin-truncated systems have been studied since the early 50’s in fluid mechanics [T.D. Lee, Quart Appl Math 10, 69 (1952)]. These spatially-periodic classical ideal fluids are known to admit, when spectrally truncated at wavenumber kmaxkmax, absolute equilibrium solutions with Gaussian statistics and equipartition of kinetic energy among all Fourier modes.

The scaling of the correlation time around thermal equilibrium for the 1D Galerkin-truncated Burgers equation with equilibrium forcing is shown to display, as expected, a crossover from Edwards-Wilkinson to KPZ scaling when forcing and dissipation are jointly decreased. A new crossover to a third regime is characterized in the inviscid limit of vanishing forcing and dissipation.

12:45 to 14:00 -- Lunch
14:00 to 15:00 Rahul Pandit Universal properties of the spatiotemporally chaotic state of the one-dimensional Kuramoto-Sivashinsky equation

The spatiotemporally chaotic state of the one-dimensional Kuramoto-Sivashinsky (KS) equation is often characterised by height correlations. There is compelling numerical evidence that the long-distance and long-time behaviours of these height correlation functions is in the Kardar-Parisi-Zhang (KPZ) universality class. We use extensive direct numerical simulations to show that this spatiotemporally chaotic state of the KS equation also displays Tracy-Widom and related distributions that are now well known for the one-dimensional KPZ equation.
This work has been done with Dipankar Roy at the Indian Institute of Science

15:00 to 15:30 -- Tea/coffee break
15:30 to 16:30 Herbert Spohn The H_{XXZ} line ensemble and KPZ universality

The 2D statistical mechanics of the XXZ chain is a system of nonintersecting random walks, where Δ regulates the interaction between the lines (Δ = 0 is free fermion, Δ > 0 attractive, Δ < 0 repulsive). KPZ fluctuations are expected to show at the stochastic line and at facet edges. We discuss earlier results and explain more recent progress to establish such behavior. This is joint work with Michael Praehofer..

Thursday, 31 January 2019
Time Speaker Title Resources
10:00 to 11:00 Daniel Remenik The KPZ fixed point - (Lecture 3)

The Kardar-Parisi-Zhang (KPZ) universality class is a broad class of models coming from mathematical physics which includes random interface growth, directed random polymers, interacting particle systems, and random stirred fluids. These models share a very special and rich asymptotic fluctuation behavior, which is loosely characterized by fluctuations which grow like t^{1/3} as time t evolves, decorrelate at a spatial scale of t^{2/3}, and have certain very special limiting distibutions; this fluctuation behavior is model independent but depends on the initial data, and in some important cases it is connected with distributions coming from random matrix theory.

A somewhat vague conjecture in the field was that there should be a universal, scaling invariant limit for all models in the KPZ class, containing all the fluctuation behavior seen in the class. In these lectures I will describe joint work with K. Matetski and J. Quastel [5] where we were able to construct and give a complete description of this limiting process, known as the KPZ fixed point. This limiting universal process is a Markov process, taking values in real valued functions which look locally like Brownian motion.

The construction follows from an novel exact solution for one of the most basic models in the KPZ class, the totally asymmetric exclusion process (TASEP), for arbitrary initial condition. This formula is given as the Fredholm determinant of a kernel involving the transition probabilities of a random walk forced to hit a curve defined by the initial data, and in the KPZ 1:2:3 scaling limit the formula leads in a transparent way to a Fredholm determinant formula given in terms of analogous kernels based on Brownian motion.

Two sets of lecture notes [6,7] serve as a good complement to the mini-course.

References:

1. A. Borodin, P. L. Ferrari, M. Prähofer, and T. Sasamoto. Fluctuation properties of the TASEP with periodic initial configuration. J. Stat. Phys. 129.5-6 (2007), pp. 10551080.
2. A. Borodin, I. Corwin, and D. Remenik. Multiplicative functionals on ensembles of non-intersecting paths. Ann. Inst. H. Poincaré Probab. Statist. 51.1 (2015), pp. 28–58.
3. I. Corwin, J. Quastel, and D. Remenik. Continuum statistics of the Airy 2 process. Comm. Math. Phys. 317.2 (2013), pp. 347–362.
4. J. Quastel, D. Remenik. How flat is flat in random interface growth? To appear in Trans. AMS. arXiv:1606.09228.
5. K. Matetski, J. Quastel and D. Remenik. The KPZ fixed point. arXiv:1701.00018.
6. K. Matetski, J. Quastel. From TASEP to the KPZ fixed point. arXiv:1710.02635.
7. D. Remenik. Course notes on the KPZ fixed point.
8. T. Sasamot o. Spatial correlations of the 1D KPZ surface on a flat substrate. J. Phys. A 38.33 (2005), p. L549.
11:00 to 11:15 -- Discussions
11:15 to 11:45 -- Tea/coffee break
11:45 to 12:45 Allan Sly The Slow Bond Model with Small Perturbations

The slow bond model is the totally asymmetric simple exclusion process (TASEP) in which particles cross the edge at the origin at rate 1-\epsilon rather than at rate 1. Janowsky and Lebowitz asked if there was a global slowdown in the current for all epsilon > 0. Using a range of of theory and simulations two groups of physicists came to opposing conclusions on this question. With Basu and Sidoravicius, this was settled establishing that there is a slowdown for any positive epsilon. In the current work we illuminate reason that this problem was difficult to resolve using simulations by analysing the effect of the perturbation at epsilon tends to 0 and showing it decays faster than any polynomial. Joint work with Lingfu Zhang and Sourav Sarkar.

12:45 to 14:00 -- Lunch
15:00 to 15:30 -- Tea/coffee break
Friday, 01 February 2019
Time Speaker Title Resources
10:00 to 11:00 Daniel Remenik The KPZ fixed point - (Lecture 5)

The Kardar-Parisi-Zhang (KPZ) universality class is a broad class of models coming from mathematical physics which includes random interface growth, directed random polymers, interacting particle systems, and random stirred fluids. These models share a very special and rich asymptotic fluctuation behavior, which is loosely characterized by fluctuations which grow like t^{1/3} as time t evolves, decorrelate at a spatial scale of t^{2/3}, and have certain very special limiting distibutions; this fluctuation behavior is model independent but depends on the initial data, and in some important cases it is connected with distributions coming from random matrix theory.

A somewhat vague conjecture in the field was that there should be a universal, scaling invariant limit for all models in the KPZ class, containing all the fluctuation behavior seen in the class. In these lectures I will describe joint work with K. Matetski and J. Quastel [5] where we were able to construct and give a complete description of this limiting process, known as the KPZ fixed point. This limiting universal process is a Markov process, taking values in real valued functions which look locally like Brownian motion.

The construction follows from an novel exact solution for one of the most basic models in the KPZ class, the totally asymmetric exclusion process (TASEP), for arbitrary initial condition. This formula is given as the Fredholm determinant of a kernel involving the transition probabilities of a random walk forced to hit a curve defined by the initial data, and in the KPZ 1:2:3 scaling limit the formula leads in a transparent way to a Fredholm determinant formula given in terms of analogous kernels based on Brownian motion.

Two sets of lecture notes [6,7] serve as a good complement to the mini-course.

References:

1. A. Borodin, P. L. Ferrari, M. Prähofer, and T. Sasamoto. Fluctuation properties of the TASEP with periodic initial configuration. J. Stat. Phys. 129.5-6 (2007), pp. 10551080.
2. A. Borodin, I. Corwin, and D. Remenik. Multiplicative functionals on ensembles of non-intersecting paths. Ann. Inst. H. Poincaré Probab. Statist. 51.1 (2015), pp. 28–58.
3. I. Corwin, J. Quastel, and D. Remenik. Continuum statistics of the Airy 2 process. Comm. Math. Phys. 317.2 (2013), pp. 347–362.
4. J. Quastel, D. Remenik. How flat is flat in random interface growth? To appear in Trans. AMS. arXiv:1606.09228.
5. K. Matetski, J. Quastel and D. Remenik. The KPZ fixed point. arXiv:1701.00018.
6. K. Matetski, J. Quastel. From TASEP to the KPZ fixed point. arXiv:1710.02635.
7. D. Remenik. Course notes on the KPZ fixed point.
8. T. Sasamot o. Spatial correlations of the 1D KPZ surface on a flat substrate. J. Phys. A 38.33 (2005), p. L549.
11:00 to 11:15 -- Discussions
11:15 to 11:45 -- Tea/coffee break
11:45 to 12:45 Christopher Hoffman Bigeodesics in fist and last passage percolation

First passage percolation is a model in statistical physics of random growth. A longstanding question, due to Furstenberg, is whether there are bi-infinite geodesics. This question is of interest to physicists due to its connections with ground states of the Ising model. We will discuss recent progress on this question. We will also show that in the related model of last passage percolation that there are no non-trivial bi-infinite geodesics a.s. This is based on joint works with Daniel Ahlberg, Riddhipartum Basu and Allan Sly

12:45 to 14:00 -- Lunch
14:00 to 15:00 Vladas Sidoravicius Random walks in growing domains - recurrence vs transienc

We will discuss set of models where the random walk (or Brownian motion) moves in a restricted domain D(t), which itself evolves in time. This evolution could be independent of random walk evolution, but still affecting its motion, or evolution of D i tself could be affected by random walk.
Then I will focus  on the phase transition (recurrence vs transience) for once reinforced random walks and some other self interacting processes.

15:00 to 15:30 -- Tea/coffee break
Monday, 04 February 2019
Time Speaker Title Resources
10:00 to 11:00 Lionel Levine Abelian networks and sandpile models

In this mini-course, we explore particle systems with an “abelian property”: the order of certain interactions does not matter. Deepak Dhar [D] observed that many dynamical questions (what does this particle system do?) have a computational side (what can this network of automata compute?). The computational counterpart of the abelian property is a “least action principle”: the particles conspire to solve a certain optimization problem, by reaching stability in the most efficient possible way.

We are interested in the phase transition between activity and fixation, and in universal properties of the “threshold state” that separates the two phases. The dynamical question “will this system fixate?” corresponds to the computational question “will this program halt?”. Alan Turing proved in 1936 that the latter question is undecidable in general. Hence, we should expect these systems to be hard, and there will be no completely satisfactory general theory; but questions in the neighborhood of an undecidable question are where the most fruitful mathematics lies!

References
[BL] Bond, Levine, https://arxiv.org/abs/1309.3445
[C] Cairns, https://arxiv.org/abs/1508.00161
[D] Dhar, https://doi.org/10.1016/j.physa.2006.04.004
[H+] Holroyd et al., https://arxiv.org/abs/0801.3306
[J] Jarai, https://arxiv.org/abs/1401.0354
[L] Levine, https://arxiv.org/abs/1402.3283
[LP] Levine, Peres, https://arxiv.org/abs/1611.00411
[RSZ] Rolla,Sidoravicius,Zindy https://arxiv.org/abs/1707.06081

11:00 to 11:15 -- Discussion
11:15 to 11:45 -- Tea/coffee break
11:45 to 12:45 Rajat Subhra Hazra A PDE approach to scaling limit of random interface models on Z^d

In this talk we shall discuss the scaling limit of Gaussian interface models, where the covariance structure comes from a discrete partial differential equation. In some models of random interfaces, the explicit description of the covariance is either lacking or sometimes difficult to derive. We suggest an approach through the approximation of solutions of continuum PDEs through discrete solutions by finite difference methods. We discuss the implications of such approximation results in the cases of the discrete Gaussian free field, the Membrane model, and the mixed model containing both gradient and Laplacian interaction. We derive the weak convergence in appropriate spaces, depending on the dimension of the lattice. This talk is based on joint and on-going works with Biltu Dan (ISI, Kolkata) and Alessandra Cipriani (TU, Delft).

12:45 to 14:00 -- Lunch
14:00 to 15:00 Wioletta Ruszel Scaling limits of odometers in sandpile models
15:00 to 15:30 -- Tea/ Coffee break
15:30 to 16:30 -- Colloqium
Tuesday, 05 February 2019
Time Speaker Title Resources
10:00 to 11:00 Lionel Levine Abelian networks and sandpile models (Lecture 2)

In this mini-course, we explore particle systems with an “abelian property”: the order of certain interactions does not matter. Deepak Dhar [D] observed that many dynamical questions (what does this particle system do?) have a computational side (what can this network of automata compute?). The computational counterpart of the abelian property is a “least action principle”: the particles conspire to solve a certain optimization problem, by reaching stability in the most efficient possible way.

We are interested in the phase transition between activity and fixation, and in universal properties of the “threshold state” that separates the two phases. The dynamical question “will this system fixate?” corresponds to the computational question “will this program halt?”. Alan Turing proved in 1936 that the latter question is undecidable in general. Hence, we should expect these systems to be hard, and there will be no completely satisfactory general theory; but questions in the neighborhood of an undecidable question are where the most fruitful mathematics lies!

References
[BL] Bond, Levine, https://arxiv.org/abs/1309.3445
[C] Cairns, https://arxiv.org/abs/1508.00161
[D] Dhar, https://doi.org/10.1016/j.physa.2006.04.004
[H+] Holroyd et al., https://arxiv.org/abs/0801.3306
[J] Jarai, https://arxiv.org/abs/1401.0354
[L] Levine, https://arxiv.org/abs/1402.3283
[LP] Levine, Peres, https://arxiv.org/abs/1611.00411
[RSZ] Rolla,Sidoravicius,Zindy https://arxiv.org/abs/1707.06081

11:00 to 11:15 -- Discussion
11:15 to 11:45 -- Tea/ Coffee break
11:30 to 11:45 -- Tea/ Coffee break
11:45 to 12:45 Christopher Hoffman Frogs on Trees

The frog model is a branching random walk on a graph in which particles branch only at unvisited sites. I will discuss a collection of results that are joint with Tobias Johnson and Matthew Junge. These papers all discuss the frog model on regular trees. On infinite trees we consider the question of transience and recurrence. On finite depth trees we examine the time it takes before every vertex on the graph is visited.

12:45 to 14:00 -- Lunch
14:00 to 15:00 Vladas Sidoravicius Random walks in growing domains - recurrence vs transience

We will discuss set of models where the random walk (or Brownian motion) moves in a restricted domain D(t), which itself evolves in time. This evolution could be independent of random walk evolution, but still affecting its motion, or evolution of D itself could be affected by random walk.
Then I will focus on the phase transition (recurrence vs transience) for once reinforced random walks and some other self interacting processes.

15:00 to 15:30 -- Tea/ Coffee break
Wednesday, 06 February 2019
Time Speaker Title Resources
10:00 to 11:30 Leonardo Rolla Absorbing-State Phase Transitions

Modern statistical mechanics offers a large class of driven-dissipative stochastic systems that naturally evolve to a critical state, of which Activated Random Walks are perhaps the best example. The main pursuit in this field is to show universality of critical parameters, describe the critical behavior, the scaling relations and critical exponents of such systems, and the connection between driven-dissipative dynamics and conservative dynamics in infinite space. The study of this model was an open challenge for a long time, then it had signifi cant partial progress a decade ago, and got stuck again. Through the last 5 years it has seen exciting progress thanks to contributions by Asselah, Basu, Cabezas, Ganguly, Hoffman, Schapira, Sidoravicius, Sousi, Stauffer, Taggi, Teixeira, Tournier, Zindy, and myself. These covered most of the questions regarding existence of an absorbing and an active phase for different ranges of parameters, and current efforts are drifting towards the description of critical states, scaling limits, etc. We will summarize the current state of art and discuss some of the many open problems.

11:45 to 12:45 Deepak Dhar --
12:45 to 14:00 -- Lunch
14:00 to 15:00 Punyabrata Pradhan Hydrodynamics of conserved stochastic sandpiles

We shall discuss conserved stochastic sandpiles (CSSs), which exhibit an active-absorbing phase transition upon tuning density. We demonstrate that a broad class of CSSs possesses a remarkable hydrodynamic structure: There is an Einstein relation, which connects bulk-diffusion coefficient, conductivity, and mass fluctuation. Consequently, density large-deviations are governed by an equilibrium-like chemical potential. We also derive two scaling relations, which could help us to settle the long standing issue of universality in such systems.

15:00 to 15:30 -- Tea/ Coffee break
15:30 to 16:30 Pradeep Kumar Mohanty Universality in sandpile models

The self-organized critical (SOC) state of sndpile models can be understood as the critical state of an absorbing state phase transition (APT) occurring in fixed energy sandpiles. It is still debated, whether the most generic sandpile models belongs to the universality class of Directed percolation -the most robust universality class of APT. I will discuss this issue focusing on recent simulation results of Manna and Oslo models.

Thursday, 07 February 2019
Time Speaker Title Resources
10:00 to 11:00 Rahul Dandekar Recurrence-Transience transition and Tracy-Widom growth in the Rotor-router mode

We describe the growing patterns formed in the rotor-router model, starting from noisy initial conditions. By the detailed study of two cases, we show that: (a) the boundary of the pattern for a certain class of initial conditions displays KPZ fluctuations with a Tracy-Widom distribution, (b) by changing the amount of randomness, one can induce a transition in which the rotor-router path changes from recurrent to transient. We show that this transition falls in the 3D Anisotropic Directed Percolation universality class.

11:00 to 11:15 -- Discussion
11:15 to 11:45 -- Tea/ Coffee break
11:30 to 12:30 Alessandra Cipriani The discrete Gaussian free field on a compact manifold

In this talk we aim at defining the discrete Gaussian free field (DGFF) on a compact Riemannian manifold. Since there is no canonical grid approximation of a manifold, we construct a suitable random graph that replaces the square lattice Z^d in Euclidean space, and prove that the scaling limit of the DGFF is given by the manifold continuum Gaussian free field. Joint work with Bart van Ginkel (TU Delft).

12:45 to 14:00 -- Lunch
14:00 to 15:00 Lionel Levine Abelian networks and sandpile models (Lecture 3)
15:00 to 15:30 -- Tea/ Coffee break
Friday, 08 February 2019
Time Speaker Title Resources
10:00 to 11:00 Lionel Levine Abelian networks and sandpile models (Lecture 4)
11:00 to 11:15 -- Discussion
11:15 to 11:45 -- Tea/ Coffee break
11:45 to 12:45 Kirone Mallick Continuous-time Quantum Walks

Quantum analogs of classical random walks have been defined in quantum information theory as a useful concept to implement original algorithms. Due to interference effects, statistical properties of quantum walks can drastically differ from their classical counterparts, leading to much faster computations. In this talk, we shall discuss various statistical properties of continuoustime quantum walks on a lattice, such as: survival properties of quantum particles in the presence of traps (i.e. a quantum generalization of the Donsker-Varadhan stretched exponential law), the growth of a quantum population in the presence of a source, quantum return probabilities and Loschmidt echoes.