# Analysis, Geometry, and Dynamical Systems Seminar

## Past sessions

### Sharp convergence of Markov chains (IV)

We show how to use entropy methods to obtain sharp estimates on the convergence of finite-state Markov chains to their stationary states.

Fourth session of a minicourse of 4 sessions.

### Sharp convergence of Markov chains (III)

We show how to use entropy methods to obtain sharp estimates on the convergence of finite-state Markov chains to their stationary states.

Third session of a minicourse of 4 sessions.

### Sharp convergence of Markov chains (II)

We show how to use entropy methods to obtain sharp estimates on the convergence of finite-state Markov chains to their stationary states.

Second session of a minicourse of 4 sessions.

### Sharp convergence of Markov chains (I)

We show how to use entropy methods to obtain sharp estimate on the convergence of finite-state Markov chains to their stationary states.

First session of a minicourse of 4 sessions.

### The Mathematics of making a mess (an introduction to random walk on groups)

How many random transpositions does it take to mix up $n$ cards? This is a typical question of random walk on finite groups. The answer is $(1/2)n \log n+Cn$ and there is a sharp phase transition from order to chaos as $C$ varies. The techniques involve Fourier analysis on non-commutative groups (which I will try to explain for non specialists). As you change the group or change the walk, new analytic and algebraic tools are required. The subject has wide applications (people still shuffle cards, but there are applications in physics, chemistry,biology and computer science — even for random transpositions). Extending to compact or more general groups opens up many problems. This was the first problem where the ‘cutoff phenomenon’ was observed and this has become a healthy research area.

Projecto FCT UIDB/04459/2020.

### Hydrodynamics of the simple exclusion process in random environment: a mild solution approach

In this talk, we discuss an alternative point of view, initiated by the works of Nagy (2001) and Faggionato (2007), on the hydrodynamic limit of the exclusion process. We show how this method based on duality and a mild solution representation of the empirical measures carries over to the case of random environment - both static and dynamic. In conclusion, we discuss some recent refinements of this method which allow, for instance, to obtain tightness of the sequence of empirical measures.

Joint work with S. Floreani (TU Delft), F. Redig (TU Delft) and E. Saada (Paris V Descartes).

Projecto FCT UIDB/04459/2020.

### Noise Impact on Finite Dimensional Dynamical Systems

Dynamical systems are often subjected to noise perturbations either from external sources or from their own intrinsic uncertainties. While it is well believed that noises can have dramatic effects on the stability of a deterministic system at both local and global levels, mechanisms behind noise surviving or robust dynamics have not been well understood especially from distribution perspectives. This talk attempts to outline a mathematical theory for making a fundamental understanding of these mechanisms in white noise perturbed systems of ordinary differential equations, based on the study of stationary measures of the corresponding Fokker-Planck equations. New existence and non-existence results of stationary measures will be presented by relaxing the notion of Lyapunov functions. Limiting behaviors of stationary measures as noises vanish will be discussed in connection to important issues such as stochastic stability and bifurcations.

Projecto FCT UIDB/04459/2020.

### From the porous medium model to the porous medium equation

The aim of this seminar is to present an overview of the porous medium model and its hydrodynamic equation, the porous medium equation. We will focus on exploring the main characteristics of this equation and how we can see it from the particle system's point of view.

Projecto FCT UID/MAT/04459/2019.

### Equilibrium fluctuations for symmetric exclusion with long jumps and infinitely extended reservoirs

The aim of this work is the analysis of fluctuations around equilibrium for a diffusive and symmetric exclusion process with long jumps and infinitely extended reservoirs (introduced in [BGJO]). In particular we study how the parameters characterizing the model change the behavior of the stochastic fluctuations of the macroscopic density of particles around the equilibrium. We will see how the SPDE involved will pass from the one which solution is a generalized Ornstein-Uhlenbeck process when the reservoirs are weak, to an SPDE without diffusive term when the reservoirs are so strong that the fluctuations are only caused by their action.

Joint work with C. Bernardin, P. Gonçalves and M. Jara.

### Reference

[BGJO] Bernardin, C., Gonçalves, P. and Oviedo Jimenez, B. Slow to fast infinitely extended reservoirs for the symmetric exclusion process with long jumps.

Projecto FCT UID/MAT/04459/2019.

### KPZ universality for last passage percolation models.

In this seminar we consider last passage percolation on $\mathbb{Z}^2$, a model in the Kardar–Parisi–Zhang (KPZ) universality class. We will investigate the universality of the limit distributions of the last passage time for different settings. In the first part we analyze the correlations of two last passage times for different ending points in a neighbourhood of the characteristic. For a general class of random initial conditions, we prove the universality of the first order correction when the two observation times are close. In the second part we consider a model of last passage percolation in half-space and we obtain the distribution of the last passage time for the stationary initial condition.

Projecto FCT UID/MAT/04459/2019.

### A short KPZ story

The aim of this talk is to present a few models in the Kardar–Parisi–Zhang (KPZ) universality class, a class of stochastic growth models that have been widely studied in the last 30 years. We will focus in particular on last passage percolation (LPP) models. They provide a physical description of combinatorial problems, such as Ulam's problem, in terms of zero temperature directed polymers; but also a geometric interpretation of an interacting particle system, the totally asymmetric simple exclusion process (TASEP); and of a system of queues and servers. Moreover, in the large time limit, they share statistical features with certain ensembles of random matrices.

Projecto FCT UID/MAT/04459/2019.

### A Mini-course in large deviations (III)

Large deviations have importance and applications in different areas, as Probability, Statistics, Dynamical Systems, and Statistical Mechanics. In plain words, large deviations correspond to finding and proving the (exponentially small) probability of observing events not expected by the law of large numbers. In this mini-course we introduce the topic including a discussion on the general statement of large deviations and the some of the usual challenges when facing the upper/lower bound large deviations.

Projecto FCT UID/MAT/04459/2019.

### A Mini-course in large deviations (II)

Large deviations have importance and applications in different areas, as Probability, Statistics, Dynamical Systems, and Statistical Mechanics. In plain words, large deviations correspond to finding and proving the (exponentially small) probability of observing events not expected by the law of large numbers. In this mini-course we introduce the topic including a discussion on the general statement of large deviations and the some of the usual challenges when facing the upper/lower bound large deviations.

Projecto FCT UID/MAT/04459/2019.

### A Mini-course in large deviations (I)

Large deviations have importance and applications in different areas, as Probability, Statistics, Dynamical Systems, and Statistical Mechanics. In plain words, large deviations correspond to finding and proving the (exponentially small) probability of observing events not expected by the law of large numbers. In this mini-course we introduce the topic including a discussion on the general statement of large deviations and the some of the usual challenges when facing the upper/lowerbound large deviations.

Projecto FCT UID/MAT/04459/2019.

### The Fibonacci family of dynamical universality classes

We use the theory of nonlinear fluctuating hydrodynamics to study stochastic transport far from thermal equilibrium in terms of the dynamical structure function which is universal at low frequencies and for large times and which encodes whether transport is diffusive or anomalous. For generic one-dimensional systems we predict that transport of mass, energy and other locally conserved quantities is governed by mode-dependent dynamical universality classes with dynamical exponents $z$ which are Kepler ratios of neighboring Fibonacci numbers, starting with $z = 2$ (corresponding to a diffusive mode) or $z = 3/2$ (Kardar-Parisi-Zhang (KPZ) mode). If neither a diffusive nor a KPZ mode are present, all modes have as dynamical exponent the golden mean $z=(1+\sqrt 5)/2$. The universal scaling functions of the higher Fibonacci modes are Lévy distributions. The theoretical predictions are confirmed by Monte-Carlo simulations of a three-lane asymmetric simple exclusion process.

Projecto FCT UID/MAT/04459/2019.

### Generalisations to Multispecies (V)

Second class particle; multispecies exclusion process; hierarchical matrix solution and proof; queueing interpretation.

Continuation of Lecture 4.

Projecto FCT UID/MAT/04459/2019.

### Generalisations to Multispecies (IV)

Second class particle; multispecies exclusion process; hierarchical matrix solution and proof; queueing interpretation.

Projecto FCT UID/MAT/04459/2019.

### Phase Diagram (III)

Complex zeros of nonequilibrium partition function; open ASEP phase transitions; continuous and discontinuous transitions; coexistence line.

Projecto FCT UID/MAT/04459/2019.

### Matrix Product Solution (II)

Matrix product ansatz; proof of stationarity; computation of partition function $Z_L$; large $L$ asymptotics of $Z_L$; current and density profile; combinatorial approaches.

Projecto FCT UID/MAT/04459/2019.

### Open Boundary ASEP (I)

The asymmetric simple exclusion process (ASEP) has been studied in probability theory since Spitzer in 1970. Remarkably a version with open boundaries had already been introduced as a model for RNA translation in 1968. This “open ASEP” has since the 1990’s been widely studied in the theoretical physics community as a model of a nonequilibrium system, which sustains a stationary current. In these lectures I will introduce and motivate the model then present a construction — the matrix product ansatz — which yields the exact stationary state for all system sizes. I will derive the phase diagram and analyse the nonequilibrium phase transitions. Finally I will discuss how the approach generalises to multispecies systems.

In this first lecture I will introduce the motivations; correlation functions; mean-field theory and hydrodynamic limit; dynamical mean-field theory; domain wall theory.

Projecto FCT UID/MAT/04459/2019.

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