Week 1 : 14 January -18 January

Mini-courses : Charles Bordenave

1. 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 $n^{-1/2}$.

2.

3. 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.

4.

5. 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.

6.

7. 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.

Authors: Arijit Chakrabarty, Rajat Subhra Hazra, Frank den Hollander, and Matteo Sfragara

8.

9. 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.

10.

11. 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.

13. 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.

14.

15. 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.

16.

17. 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.

18.

19. 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.

20.

21. David T 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.

22.

23. 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.

24.

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)

26.

Week 2 : 21 January -25 January
Mini-courses : Tomohiro Sasamoto
27. 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).

28.

29. 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.

30.

31. Shirshendu Ganguly

Fractal Properties of Coupled Polymer Weight Profiles via Coalescence of Geodesics

In this talk we will discuss initial outcomes of a program to understand fractal properties of coupled polymer weight profiles in (1+1)-dimensional Brownian Last Passage Percolation, based on coalescence of geodesics, the Brownian Gibbs property and other geometric tools. As a consequence, one obtains results about the the so called Airy Sheet A(x, s, y, t): a four parameter random field conjectured to be the scaling limit of polymer weights between (x, s) and (y, t) which has recently been constructed.

32.

33. 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.

34.

35. 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.

36.

37. 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).

38.

39. 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.

40. Lecture Notes

41. 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.

42.

Short Talks :

43. Terrence George

Spectra of biperiodic planar networks

A biperiodic planar network is a pair (G, c) where G is a graph embedded on the torus and c is a function from the edges of G to non-zero complex numbers. Associated to the discrete Laplacian on a biperiodic planar network is a pair (C, S) called the spectral data, where C is a curve and S is a divisor on it. We give a complete classification of networks (modulo a natural equivalence) in terms of their spectral data. The space of networks has a large group of automorphisms arising from the Y − ∆ transformation, each of which gives a discrete integrable system. In particular, the cube recurrence arises in this way. We show that these automorphisms have a simple interpretation as translations in the spectral data, leading to an explicit description of the automorphism group.

44.

45. Matteo Mucciconi

Stationary Higher Spin Six Vertex Model

The Higher Spin Six Vertex Model is a stochastic integrable system introduced in 2016 by Corwin and Petrov and that generalizes a number of other exactly solvable models in the KPZ Universality class, including the Six Vertex Model, the TASEP and its q-deformations and the ASEP. In this talk we discuss some geometrical properties of the stationary state of the Higher Spin Six Vertex Model and we confirm Baik-Rains fluctuations for the stationary Height functions. The talk is based on a collaboration with T. Imamura and T. Sasamoto.

46.

47. Dipankar Roy

The exact phase diagrams for a class of left-permeable asymmetric exclusion process

Asymmetric simple exclusion processes (ASEP) with multiple species are important models in non-equilibrium statistical physics. In this work, we study an integrable two species partially asymmetric exclusion process called the left-permeable ASEP or LPASEP and its multispecies generalization, called the mLPASEP. In both these models, the left boundary is permeable to all species but the right boundary is impermeable to some of them. For the former, we construct a matrix product solution for the stationary state and thereby compute the exact stationary phase diagram for densities and currents. For the latter, using projections onto the LPASEP, we derive densities and currents as well as the phase diagram in the steady state. One observes the phenomenon of dynamical expulsion of one or more species in most of the phases. We explain this phenomenon and the density profiles in each phase using interacting shocks. This is joint work with Prof. Arvind Ayyer (Indian Institute of Science Bangalore) and Dr Caley Finn (The University of Melbourne).

48.

49. Tapas Singha

Clustering, intermittency, and scaling for passive particles on fluctuating surfaces

I will talk about a scaling approach successfully characterizes clustering and intermittency in space and time, in systems of noninteracting particles driven by fluctuating surfaces. We study both the steady state and the approach to it, for passive particles sliding on one-dimensional Edwards-Wilkinson or Kardar-Parisi-Zhang (KPZ) surfaces, with particles moving either along (advection) or against (antiadvection) the growth direction in the latter case. Extensive numerical simulations are supplemented by analytical results for a sticky slider model in which particles coalesce when they meet. Results for single-particle displacement versus time show to what extent particle dynamics is slaved to the surface, while scaling properties of the probability distribution of the separation of two particles determine the scaling form of average overlap of a pair of trajectories. For the many-particle system, clustering in steady state is studied via moments of particle number fluctuations in a single stretch, revealing different degrees of spatial multiscaling with different drivings.
Temporal intermittency in steady state is established for all the three drivings, but with different exponents, reflecting strongest clustering for KPZ advection and weakest for KPZ antiadvection. Finally, we consider the approach to the steady state and study both the flatness and the evolution of equal-time correlation functions as in coarsening of phase- ordering systems. Our studies give clear evidence for a simple scaling description of the approach to steady state, with the scale set by a length that grows in time. An investigation of aging properties reveals that flatness is nonmonotonic in time with two distinct branches and that a scaling description holds for each one.

50.

51. Lingfu Zhang

Convergence of empirical distributions in exponential last passage percolation

In the 2D exponential last passage percolation, the geodesic from (0, 0) to (n, n) is the up-right path with the maximum sum of the random variables along the path. We study the very local behavior of the geodesic. For every point in the geodesic, we take a k×k box centered around it, and consider the law of the empirical distribution of the boxes along the geodesic. We show that it converges in probability as n → ∞. The main idea of the proof is to show that the very local behavior around different points in the geodesic are asymptotically i.i.d. The KPZ fixed point formula and other observations of the exact solvable model are widely used.
This is joint work with Allan Sly.

52.

Week 3 : 28 January - 1 February
Mini-courses : Daniel Remenik

53. 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 $k_{max}$, 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.

54.

55. 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.

56.

57. 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.

58.

59. 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

60.

61. 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.

62.

63. 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

64.

65. 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.

66.

67. 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.

68.

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.

70.

71. 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.

72.

73. 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..

74.

Week 4: 04 February - 08 February
Mini-courses : Lionel Levine

75. 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).

76.

77. 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.

78.

79. Deepak Dhar

The active-absorbing state transition in the fixed energy sand-pile models

The active absorbing state transition in the fixed energy sand-pile mod els has been a topic of much interest. In these model, the steady state shows a transition from the inactive to active phase, as the density of particles is increased. The relationship of this transition to the directed percolation universality class, or the conserved directed percolation class has been a topic of some controversy. I will discuss our recent work on the directed Oslo model, which shows different critical behaviors for different parameter ranges. I will also discuss the equivalence of this model to a generalization of the Edwards -Wilkinson model of interface growth with additive but non-linear noise.

80.

81. Rajat 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).

82.

83. 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.

84.

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.

86.

87. 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.

88.

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.

90.

91. 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, Richey, 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.

92.

93. Wioletta Ruszel

Scaling limits of odometers in sandpile models

The divisible sandpile model is a special case of the class of continuous sandpile models on a graph V where the initial configuration is random and the evolution deterministic. Under certain conditions on the initial configuration the model will stabilize to the all 1 configuration. The amount of mass (u(x))_{x\in V} that is emitted from x \in V during stabilization is called the odometer. Depending on the initial configuration and the way how mass is distributed one can show that the scaling limit of u can be either a fractional Gaussian field or alpha-stable.
The results presented in this talk are joint work with L.Chiarini (IMPA/TU Delft), A. Cipriani (TU Delft), J. de Graaff (TU Delft), M. Jara (IMPA) and R. Hazra (ISI Kolkatta).

94.

Mathematics of Multi-particle diffusion limited aggregation

In 1980 H. Rosenstock and C. Marquardt introduced the following stochastic aggregation model on Z d : Start with particles distributed according to the product Bernoulli measure with parameter 0 < µ < 1, conditioned not to have particle at the origin. At the origin we place a special particle, which is called the ”seed of aggregate”. Non-aggregated particles move as continuous-time simple random walks obeying the exclusion rule. The aggregate grows by attaching particles to its surface whenever a particle attempts to jump onto it. Aggregated particles do not move. This evolution is called Multi-particle Diffusion Limited Aggregation (MDLA). MDLA model even in dimension 1 has highly nontrivial behavior: If exclusion rule is removed and non-aggregated particles move as simple symmetric random walks (in this case the initial density is product of Poisson with parameter 0 < ρ, H. Kesten and V. Sidoravicius (2008) proved that if ρ < 1, then the aggregate is growing sub-linearly, with exponent 1/2, and A. Sly (2016) showed that if ρ > 1 it advances linearly, establishing a phase transition.

In my talk I will briefly review known results and will focus on the progress in dimensions d ≥ 2. Our main result (joint with A. Stauffer) states that for the exclusion version of the process if d > 1 and the initial density of particles is close enough to 1, then with positive probability the aggregate has linearly growing arms; that is, there exists a constant c > 0 so that at time t the aggregate contains a point at distance at least ct from the origin, for all t, and also it obeys certain type of the shape theorem. The key conceptual element of our analysis is the introduction and study of a new FPP type growth process. Study of this process indicates that high density MDLA may belong to KPZ universality class.

96.

97. Debleena Thacker

Border Aggregation Model

Start with a graph with a subset of vertices called the border. A particle released from the origin performs a random walk on the graph until it comes to the immediate neighbourhood of the border, at which point it joins this subset thus increasing the border by one point. Then a new particle is released from the origin and the process repeats until the origin becomes a part of the border itself. We are interested in the total number ξ of particles to be released by this final moment. We show that this model covers OK Corral model as well as the erosion model, and obtain distributions and bounds for ξ in cases where the graph is star graph, regular tree, and a ddimensional lattice.

Levine and Peres (2007) observed that the border aggregation model on ddimensional lattice can be considered as an inversion of the classical diffusion-limited-aggregation model (DLA). We strengthen the bounds obtained in Kesten (1987) for DLA model to obtain a lower bound for ξ.