We consider independent Hermitian heavy-tailed random matrices. Our model includes the Lévy matrices as well as sparse random matrices with O(1) non-zero entries per row. By representing these matrices as weighted graphs, we derive a large deviation principle for key macroscopic observables. Specifically, we focus on the empirical distribution of eigenvalues, the joint neighborhood distribution, and the joint traffic distribution. As an application, we define a notion of microstates entropy for traffic distribution which is additive under free traffic convolution.[-]

Engineering analysis should match an underlying designed shape and not restrict the quality of the shape. I.e. one would like finite elements matching the geometric space optimized for generically good shape. Since the 1980s, classic tensor-product splines have been used both to define good shape geometry and analysis functions (finite elements) on the geometry. Polyhedral-net splines (PnS) generalize tensor-product splines by allowing additional control net patterns required for free-form surfaces: isotropic patterns, such as n quads surrounding a vertex, an n-gon surrounded by quads, polar configurations where many triangles join, and preferred direction patterns, that adjust parameter line density, such as T-junctions. PnS2 generalize C1 bi-2 splines, generate C1 surfaces and can be output bi-3 Bezier pieces. There are two instances of PnS2 in the public domain: a Blender add-on and a ToMS distribution with output in several formats. PnS3 generalize C2 bi-3 splines for high-end design. PnS generalize the use of higher-order isoparametric approach from tensor-product splines. A web interface offers solving elliptic PDEs on PnS2 surfaces and using PnS2 finite elements.[-]

Dans cet exposé, nous introduirons certaines chaînes de Markov simples, dites “montantes-descendantes”, sur les permutations et les graphes. Une étape de la chaîne consiste à dupliquer un élément aléatoire de la permutation ou un sommet aléatoire du graphe (pas montant), puis à supprimer un autre élément/sommet aléatoire (pas descendant). Nous prouvons que ces chaînes convergent dans la limite des grandes tailles et après renormalisation du temps vers une diffusion de Feller sur l'espace des permutons et des graphons, respectivement. Nous obtenons également une formule explicite pour la distance de séparation entre la distribution des chaînes après n pas, excluant l'apparition d'un phénomène de “cut-off”. Notre approche fonctionne dans un cadre plus général : il est basé sur des relations de commutation entre les opérateurs des pas montants et descendants, et s'inspire des travaux de Fulman, Olshanski et Borodin–Olshanski sur l'espace des partitions et le simplex de Thoma. Je ne supposerai aucune connaissance préalable des permutons, graphons, diffusions de Feller, distances de séparation, seuils, ... Travail joint (et encore en cours) avec Kelvin Rivera-Lopez, Gonzaga University.[-]

In these three lectures on quantum information and complexity, we will (1) review the basic concepts of quantum information processing units (QPUs), (2) prove a version of the claim that almost all quantum circuits are very complex in the sense that they are exponentially expensive to realize in the quantum circuit model of computation and (3) that the quantum complexity of a random quantum circuit grows linearly with the size of the circuit up to exponentially large circuits.

The underlying proof technique uses a versatile proof strategy from high-dimensional probability theory that can (and has been) readily extended to other problems within quantum computing theory and beyond.[-]

In these three lectures on quantum information and complexity, we will (1) review the basic concepts of quantum information processing units (QPUs), (2) prove a version of the claim that almost all quantum circuits are very complex in the sense that they are exponentially expensive to realize in the quantum circuit model of computation and (3) that the quantum complexity of a random quantum circuit grows linearly with the size of the circuit up to exponentially large circuits.

The underlying proof technique uses a versatile proof strategy from high-dimensional probability theory that can (and has been) readily extended to other problems within quantum computing theory and beyond.[-]

In these three lectures on quantum information and complexity, we will (1) review the basic concepts of quantum information processing units (QPUs), (2) prove a version of the claim that almost all quantum circuits are very complex in the sense that they are exponentially expensive to realize in the quantum circuit model of computation and (3) that the quantum complexity of a random quantum circuit grows linearly with the size of the circuit up to exponentially large circuits.

The underlying proof technique uses a versatile proof strategy from high-dimensional probability theory that can (and has been) readily extended to other problems within quantum computing theory and beyond.[-]

The propagation of chaos phenomenon states roughly that a large system of weakly interacting particles will remain approximately independent for all times if initialized as such. This can be quantified in terms of the distance between low-dimensional marginal distributions and suitably chosen product measures. This talk will discuss some recent sharp quantitative results of this nature, both for classical mean field diffusions and for more recently studied non-exchangeable models. These results are driven by a new "local" relative entropy method, in which low-dimensional marginals are estimated iteratively by adding one coordinate at a time, leading to surprising improvements on prior results obtained by "global" arguments such as subadditivity inequalities. In the non-exchangeable setting, we exploit a surprising connection with first-passage percolation.[-]

These lectures review the physical and mathematical foundations of the phenomenon of spontaneous stochasticity. The essence is that stochastic classical equations and/or with randomness in their initial data, when considered along a sequence where the randomness vanishes but also the dynamics becomes singular, can have limits described by a probability measure over the non-unique weak solutions of the limiting deterministic dynamics with deterministic initial data. Furthermore, the limiting probability measure is often universal, independent of the precise sequence considered, so that the stochastic limit is then the well-posed solution of the Cauchy problem for the limiting deterministic dynamics. In the firstlecture, we discuss Lagrangian spontaneous stochasticity, which has its origin in the 1926 paper of Lewis Fry Richardson on turbulent 2-particle dispersion. As first realized by Krzysztof Gawędzki and collaborators in 1997, Lagrangian spontaneous stochasticity is necessary for anomalous dissipation of a scalar advected by a turbulent fluid flow. In the second lecture, we discuss Eulerian spontaneous stochasticity, which was anticipated in the 1969 work of Edward Lorenz on predictability of turbulent flows. After the convex integration studies of De Lellis, Székelyhidi, and others showed that Euler equations with suitable initial data may admit infinitely many, non-unique admissible weak solutions, it became clear that Lorenz' pioneering work could be understood in the framework of spontaneous stochasticity. Finally, in the third lecture we discuss outstanding problems and more recent work on spontaneous stochasticity, both Lagrangian and Eulerian. We focus in particular on statistical-mechanical analogies, on the chaotic dynamical properties necessary to achieve universality,on the use of renormalization group methods to calculate spontaneous statistics in dynamics with scale symmetries, and finally on the challenge of observing spontaneous stochasticity in laboratory experiments.[-]

Even before the introduction of Conformal Field Theory by Belavin, Polyakov and Zamolodchikov, it appeared indirectly in the work of den Nijs and Nienhuis using Coulomb gas techniques. The latter postulate (unrigorously) that height functions of lattice models of statistical mechanics (like percolation, Ising, 6-vertex models etc) converge to the Gaussian Free Field, allowing to derive many exponents and dimensions.This convergence remains in many ways mysterious, in particular it was never formulated in the presence of a boundary, but rather on a torus or a cylinder. We will discuss the original arguments as well as some recent progress, including possible formulations on general domains or Riemann surfaces and their relations to CFT, SLE and conformal invariance of critical lattice models. Interestingly, new objects in complex geometry and potential theory seem to arise.[-]

An $N$ dimer cover of a graph is a collection of edges such that every vertex is contained in exactly $N$ edges of the collection. The multinomial dimer model studies a family of natural but non-uniform measures on $N$ dimer covers. In the large $N$ limit, this model turns out to be exactly solvable in a strong sense, in any dimension $N$. In this talk, I will define the model, and discuss its properties on subgraphs of lattices in the iterated limit as the multiplicity $N$ and then the size of the graph go to infinity, analogous to the scaling limit question for 2D standard dimers addressed by Cohn, Kenyon, and Propp. In this setting we can explicitly compute limit shapes in some examples, in particular for the Aztec diamond and a 3D analog called the Aztec cuboid. I will also discuss the surrounding theory, including explicit formulas for the free energy, large deviations, EulerLagrange equations, gauge functions, and regularity properties of limit shapes.This is joint work with Rick Kenyon.[-]