## Tentative Schedule

** Mo, 10.12.2018 **

arrival day

Tu, 11.12. | We, 12.12. | Fr, 14.12. | Sa, 15.12. | |

8:00-8:55 | Registration | |||

8:55-9:00 | Welcoming | |||

chair: |
A. Klein |
J. Fontbona |
M. Cabezas |
G. Moreno |

9:00-9:50 | L. Erdös I | A. Elgart I | B. Virag II | A. Elgart III |

10:00-10:30 | coffee |
coffee |
coffee |
coffee |

10:30-11:20 | D. Remenik I | B. Virag I | D. Remenik II | L. Erdös III |

11:30-12:20 | L. Erdös II | A. Elgart II | B. Virag III | D. Remenik III |

12:30-14:00 | lunch |
lunch |
lunch |
lunch |

chair: |
G. De Nittis |
C. Delplancke |
A. Taarabt |
M. Duarte |

14:30-15:10 | O. Zeitouni | K. Schnelli | W. de Siqueira | J. Ramirez |

15:15-15:55 | X. Zeng | D. Schröder | P. Müller | R. Movassagh |

16:00-16:30 | coffee |
coffee |
coffee |
coffee |

16:30-17:10 | A. Dietlein | M. Aizenman | B. Gentz | C. Rojas |

17:15-17:55 | P. Hislop | K. Khanin | J. Schenker | J. Bellissard |

**Thursday is free**,

**Conference dinner: Wednesday evening, 20:00 Parillera Don Jorge, Carlos Bories 440 ** At 19:40 there will be a gathering in the Lobby of Costaustralis for walking to the conference dinner.

## Speakers

### Mini -courses:

**Alexander Elgart (Virginia Tech)****Disordered quantum spin chains**

presentation available here,*abstract*

Quantum spin chains provide some of the mathematically most accessible examples of quantum many-body systems. However, even these toy models pose considerable analytical and numerical challenges, due to the fact that the number of degrees of freedom involved grows exponentially fast with the system’s size. Last years have seen a significant effort in physics community to understand the dynamical behavior of disordered quantum systems, in particular to provide at least phenomenological description of their quantum phases. These phases are expected to include a ‘thermal’ phase, where statistical mechanics is obeyed and a ‘many body localized’ phase, where the initial state fails to thermally equilibrate. Mathematically, the field is in its initial stage of development, even from the conceptual point of view. We will start these lectures with the brief description of the relevant physical concepts such as thermalization, exponential clustering, area law, and locality. We will then discuss recent progress in establishing many body localization in the droplet phase of the disordered XXZ chain. Finally, we will introduce new approach to many body localization that works beyond the droplet phase (joint work in progress with A. Klein)

**Laszlo Erdös (IST Austria)****Self-consistent Dyson equations and their application in random matrix theory.**

presentation available here,*abstract*

E. Wigner’s revolutionary vision postulated that the local eigenvalue statistics of large random matrices are independent of the details of the matrix ensemble apart from its basic symmetry class. There have recently been a substantial development to prove Wigner’s conjecture for larger and larger classes of matrix ensembles motivated by applications. They include matrices with entries with a general correlation structure and addition of deterministic matrices in a random relative basis. Apart from giving a concise overview of the basic methods, I will focus on the proof of the local density of states, the most model specific part of the theory. The key analytic tool is the stability properties of a sophisticated nonlinear equation, the Dyson equation.

**Daniel Remenik (U. de Chile)****The KPZ fixed point**

presentation available here,*abstract*

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 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.

**Balint Virag (U. Toronto)****The directed landscape**

presentation available here,*abstract*

This minicourse is about Brownian last passage percolation and its scaling limit we recently constructed with Dauvergne and Ortmann.

I will start with the basics and explain the full proof. The arguments are mostly elementary and geometric. Reliance on formulas will be minimal. Some highlights:

– Last passage paths converge to random continuous functions which are more regular than Brownian motion: 2/3-eps Holder. They are geodesics in the directed landscape.

– The directed landscape is a stationary independent-increment process on the metric composition semigroup. It has the famous 1-2-3 scaling.

– The increments are Airy sheets. Their statistics are contained in the Airy line ensemble (will explain all the terms).

– The directed landscape contains all the desired information in the limit. Previous constructions, such as the Airy line ensemble, the KPZ fixed point and multi-time limits are all marginals of this process.

### Talks:

**Michael Aizenman (U. Princeton)****Quenched disorder’s effects on phase transitions in low dimensions***abstract:*

As an example of the Imry-Ma phenomenon, the discontinuity of the magnetization in the two dimensional Ising model is unstable to the addition of quenched random magnetic field of uniform variance, even if that is small. The talk will focus on a quantitative version of the statement, yielding a power-law upper bound on the decay rate of the effect of boundary conditions on the magnetization in finite systems. At sufficiently high disorder the decay is actually exponentially fast. We shall discuss the still open question whether that persists through arbitrarily weak disorder.

The analysis proceeds through a better quantified variant of the Aiz.-Wehr proof of the Imry-Ma rounding effect. (Joint work with Ron Peled)

**Jean Bellissard (WMU Münster / Georgia Tech)****Progress on spectrum approximation for Schrödinger Operators with aperiodic potentials**

presentation available here,*abstract:*

In collaboration with Siegfried Beckus and Giuseppe De Nittis, this program intends to provide a new very general method, to compute the spectrum of Schrödinger operators with aperiodic potentials. Results concern only the spectrum as a set, with very few results on the spectral measure. The method is independent of the dimension. It consists in approximating the potential by periodc ones. Since 2016, several results have been published or posted online and this talk will give a taste of what they are.

**Adrian Dietlein (LMU München)****Poisson local eigenvalue statistics for continuum random Schrödinger operators**

presentation available here,*abstract:*

Poissonian local eigenvalue statistics are believed to be a characteristic feature of spectrally localized quantum mechanical systems. For localized random Schrödinger operators Poissonian level statistics have however only been proven for the lattice Anderson model and close relatives: The proof of a key ingredient, the Minami estimate, crucially relied on the rank-1 character of the single-site potential. We present a more flexible approach towards Minami’s estimate, which for instance works at the bottom of the spectrum of a continuum random Schrödinger operator with sufficiently regular single-site distributions. The talk is based on joint work with Alex Elgart.

**Barbara Gentz (U. Bielefeld)****Noise-induced synchronization in circulant networks of weakly coupled commensurate oscillators**

presentation available here,*abstract:*

We will discuss synchronization in circulant networks of commensurate harmonic oscillators. In such networks, weak multiplicative-noise coupling can amplify some of the system’s eigenmodes and thus lead to asymptotic eigenmode synchronization. The Euler–Fermat theorem allows to relate a class of circulant noise-coupling topologies to their induced synchronization patterns. Critical numbers of oscillators at which these synchronization patterns change can then be identified.

These synchronization results rely on an averaging principle for an outer-product process which captures all of the uncoupled system’s first integrals.

**Peter Hislop (U. Kentucky)****Dependence of the density of states on the probability distribution for random Schrödinger operators**

presentation available here,*abstract:*

I will discuss joint work with C. A. Marx on the density of states measure (DOSm) for random Schrödinger operators. We prove that the DOSm is Hölder-continuous in the single-site probability measure ν and provide quantitative estimates on the modulus of continuity. The framework we develop is general enough to extend to a wide range of discrete and continuous random operators. Additional results include quantitative continuity estimates on dependence of the integrated density of states (IDS) single-site probability measure, as well as quantitative continuity estimates for the disorder dependence of the DOSm and the IDS in the weak disorder regime. These results hold for rather general single-site probability measures. As a further application of the main result, we establish quantitative continuity results for the Lyapunov exponent of random Schrödinger operators for d=1 in the probability measure with respect to the weak-* topology.

**Konstantin Khanin (U. Toronto)****On stationary solutions to the stochastic heat equation**

presentation available here,*abstract:*

I shall discuss a problem of stationary solutions to the random Hamilton-Jacobi/Burgers equations. I shall also present a recent result, joint with Tobias Hurth and Beatriz Navarro, on a related problem of stationary solutions to the stochastic heat equation in the case of weak disorder (D>2, small coupling constants).

**Peter Müller (LMU München)****An enhanced area law for the entanglement entropy in the random dimer model***abstract:*

We consider the random dimer model in one space dimension with Bernoulli disorder. For sufficiently small disorder, we show that the entanglement entropy exhibits (at least) a logarithmically enhanced area law if the Fermi energy coincides with a critical energy of the model where the localisation length diverges. The talk is based on joint work with Ruth Schulte.

**Jose A. Ramirez Gonzalez (U. Costa Rica)****Transitions between the hard and the soft edge in beta ensembles***abstract:*

A class of log gas models introduced by Claeys and Kuijlaars interpolates between the two classical edge regimes of random matrices. Associated to those models we present a family of random matrices that allow for generalization. Furthermore this point of view leads naturally to the use of methods from the realm of stochastic differential equations to analyze the models. Descriptions of the limiting spectra (when the matrices grow) are given in terms of stochastic integral operators. This is work in progress with B. Rider.

**Constanza Rojas Molina (U. Düsseldorf)****Random Schrödinger Operators arising in aperiodic media**

presentation available here,*abstract:*

In this talk we discuss how random Schrödinger operators appear naturally as auxiliary models to study Anderson localization in aperiodic media. We work within the framework of Delone operators, which we associate to Anderson-type operators with Bernoulli random variables, and discuss the proof of localization, and its consequences on the spectral type of Delone operators near the bottom of the spectrum. This is joint work with Peter Müller (Munich).

**Ramis Movassagh (IBM research)****Hamiltonian density of states from free probability theory: Anderson model, Floquet systems, and Quantum Spin Chains***abstract:*

Suppose the eigenvalue distributions of two matrices M_1 and M_2 are known. What is the eigenvalue distribution of the sum M_1+M_2? This problem has a rich pure mathematics history dating back to H. Weyl (1912) with many applications in various fields. Free probability theory (FPT) answers this question under certain conditions, which often involves some degree of randomness (disorder). We will describe FPT and show examples of its powers for approximating physical quantities such as the density of states of the Anderson model, quantum spin chains, and gapped vs. gapless phases of some Floquet systems. These physical quantities are often hard to compute exactly. Nevertheless, using FPT and other ideas from random matrix theory excellent approximations can be obtained. Besides the applications presented, we believe the techniques will find new applications in fresh new contexts.

**Jeffrey Schenker (Michigan State)****How big is a lattice point? (or random walks and applied chemical ecology)**

presentation available here,*abstract:*

Suppose we wish to compute the probability for a random walk to hit a particular set T within a given time. It is natural to approximate this hitting probability by the probability for Brownian motion to hit some set T’ over the same time. But which set T’ should we use? For walks in two dimensions this is a subtle problem because the probability for Brownian motion to eventually hit any disk is one, regardless of the radius. I will present recent work, based on spectral theory and ideas from renormalization theory, which provides an exact analysis of certain lattice models to compute “effective Brownian radius” for a lattice point, which can be used to obtain the best approximation of random walk hitting probabilities. I will also present strong numerical evidence that the approximation works extremely well. Finally, I will discuss the application of these notions to data analysis related to monitoring populations of agricultural pests. This talk is based on joint work with A. Becerra, T. Weicht, Z. Tilocco (math students) and J. Miller, C. Adams (MSU Entomology).

**Kevin Schnelli (KTH Stockholm)****Local law of addition of random matrices on optimal scale**

presentation available here,*abstract:*

Describing the eigenvalue distribution of the sum of two general Hermitian matrices is basic question going back to Weyl. If the matrices have high dimensionality and are in general position in the sense that one of them is conjugated by a random Haar unitary matrix, the eigenvalue distribution of the sum is given by the free additive convolution of the respective spectral distributions. This result was obtained by Voiculescu on the macroscopic scale. In this talk, I show that it holds on the microscopic scale all the way down to the eigenvalue spacing in the bulk and at the regular edges. This shows a remarkable rigidity phenomenon for the eigenvalues. Joint work with Z.G. Bao and L. Erdös.

**Dominik Schröder (IST Austria)****Cusp Universality for Wigner-type Random Matrices**

presentation available here,*abstract:*

For Wigner-type matrices, i.e. Hermitian random matrices with independent, not necessarily identically distributed entries above the diagonal, we show that at any cusp singularity of the limiting eigenvalue distribution the local eigenvalue statistics are universal and form a Pearcey process. Since the density of states typically exhibits only square root or cubic root cusp singularities, our work complements previous results on the bulk and edge universality and it thus completes the resolution of the Wigner-Dyson-Mehta universality conjecture for the last remaining universality type.

**Walter de Siqueira (U. Sao Paulo)****Large Deviations for Weakly Interacting Fermions at Equilibrium – Generating Functions as Berezin Integrals.***abstract:*

We prove that the Gaertner-Ellis generating functions associated with KMS states of weakly interacting fermions on the lattice can be written as the limit of logarithms of Gaussian Berezin integrals. The covariances of the Gaussian integrals are shown to have a uniform Pfaffian bound and to be summable in general cases of interest (including systems that are not translation invariant). The Berezin integral representation can thus be used to obtain convergent expansions of the generating function in terms of powers of its parameter. Because the results are uniform w.r.t. the free part of the interaction, they are relevant for the study of equilibrium correlations of weakly interacting fermions in random media (background potentials). In this context, a recent application on the classical behavior of the electric conductivity at microscopic scales will be discussed.

**Xiaolin Zeng (U. Strasbourg)****Three aspects on the edge reinforced random walk**

presentation available here,*abstract:*

The edge reinforced random walk is a self-interacting process, in which the random walker prefer visited edges with a bias proportional to the number of times the edges were visited. We will gently introduced the model, then present its three different points of view. Namely, as a random walk in random environment, a random operator, and a spin system. If time allows, some consequences of changing the points of view will be discussed.

**Ofer Zeitouni (Weizmann Institute)****Heat kernel for Liouville Brownian motion and the geometry of the Gaussian Multiplicative chaos***abstract:*

I will discuss recent progress in the study of heat kernels for the LBM, and their relations to distances for the GMC. Joint work with Ding and Zhang.