# Seminário de Análise, Geometria e Sistemas Dinâmicos

## Sessões anteriores

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### Generalisations to Multispecies (V)

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

Continuation of Lecture 4.

### Generalisations to Multispecies (IV)

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

### Phase Diagram (III)

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

### 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.

### 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.

### Random walks, electric networks, moving particle lemma, and hydrodynamic limits

While the title of my talk is a riff on the famous monograph Random walks and electric networks by Doyle and Snell, the contents of my talk are very much inspired by the book. I'll discuss how the concept of electrical resistance can be applied to the analysis of interacting particle systems on a weighted graph. I will start by summarizing the results of Caputo-Liggett-Richthammer, myself, and Hermon-Salez connecting the many-particle stochastic process to the one-particle random walk process on the level of Dirichlet forms. Then I will explain how to use this type of energy inequality to bound the cost of transporting particles by the effective resistance, and to perform coarse-graining on a class of state spaces which are bounded in the resistance metric. This new method plays a crucial role in the proofs of scaling limits of boundary-driven exclusion processes on the Sierpinski gasket.

### On the algebraic solvability of the MPA approach to the Multispecies SSEP

On this mini-course we will learn how to extend the MPA formulation to the multispecies case with the very simple SSEP dynamics. Due to the jump symmetries of each particle in the bulk, this formulation allows us to compute any phisical quantity without resorting to the matrices representation. Nevertheless, their existence is still a necessary condition for the formulation to work. We will give an example of such matrices and focus on the exploitation of the induced algebra.

### Least energy solutions of Hamiltonian elliptic systems with Neumann boundary conditions

In this talk, we will discuss existence, regularity, and qualitative properties of solutions to the Hamiltonian elliptic system $$-\Delta u = |v|^{q-1} v\ \ \ \text{in} \ \Omega,\quad -\Delta v = |u|^{p-1} u\ \ \ \text{in} \ \Omega,\quad \partial_\nu u=\partial_\nu v=0\ \ \ \text{on} \ \partial\Omega,$$with $\Omega\subset \mathbb R^N$ bounded, both in the sublinear $pq< 1$ and superlinear $pq>1$ problems, in the subcritical regime. In balls and annuli, we show that least energy solutions are not radial functions, but only partially symmetric (namely foliated Schwarz symmetric). A key element in the proof is a new $L^t$-norm-preserving transformation, which combines a suitable flipping with a decreasing rearrangement. This combination allows us to treat annular domains, sign-changing functions, and Neumann problems, which are nonstandard settings to use rearrangements and symmetrizations. Our theorems also apply to the scalar associated model, where our approach provides new results as well as alternative proofs of known facts.

### Matrix product ansatz for the totally asymmetric exclusion process

Generally, it is very difficult to compute nonequilibrium stationary states of a particle system. It turns out that, in some cases, you can find a solution with a quite interesting structure. The goal of this first part of the seminar is to present the structure of this solution — known as matrix product solution (or matrix product ansatz) — using the totally asymmetric exclusion process (TASEP) as a toy model.

### The spectral gap of the interchange process: a review

Aldous’ spectral gap conjecture asserted that on any graph the random walk process and the interchange process have the same spectral gap. In this talk I will review the work in collaboration with T. M. Liggett and T. Richthammer from 2009, in which we proved the conjecture by means of a recursive strategy. The main idea, inspired by electric network reduction, was to reduce the problem to the proof of a new comparison inequality between certain weighted graphs, which we referred to as the octopus inequality. The proof of the latter inequality is based on suitable closed decompositions of the associated matrices indexed by permutations. I will first survey the problem, with background and consequences of the result, and then discuss the recursive approach based on network reduction together with some sketch of the proof. I will also present a more general, yet unproven conjecture.

### Mixing time of the adjacent walk on the simplex

By viewing the $N$-simplex as the set of positions of $N-1$ ordered particles on the unit interval, the adjacent walk is the continuous time Markov chain obtained by updating independently at rate $1$ the position of each particle with a sample from the uniform distribution over the interval given by the two particles adjacent to it. We determine its spectral gap and mixing time and show that the total variation distance to the uniform distribution displays a cutoff phenomenon. The results are extended to a family of log-concave distributions obtained by replacing the uniform sampling by a symmetric Beta distribution. This is joint work with Cyril Labbe' and Hubert Lacoin.

### Large-$N$ Segal-Bargmann transform with application to random matrices

I will describe the Segal-Bargmann transform for compact Liegroups, with emphasis on the case of the unitary group $U(N)$. In this case, the transform is a unitary map from the space of $L^2$ functions on $U(N)$ to the space of $L^2$ holomorphic functions on the "complexified" group $\operatorname{GL}(N;\mathbb{C})$. I will then discuss what happens in the limit as $N$ tends to infinity. Finally, I will describe an application to the eigenvalues of random matrices in $\operatorname{GL}(N;\mathbb{C})$. The talk will be self-contained and have lots of pictures.

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