Aizenman-Wehr argument for a class of disordered gradient models

We consider random gradient fields with disorder where the interaction potential $V_e$ on an edge $e$ can be expressed as $e^{-V_e(s)} = \int \rho(\mathrm{d}\kappa)\, e^{-\kappa \xi_e} e^{-\frac{\kappa s^2}{2}}$. Here $\rho$ denotes a measure with compact support in $(0,\infty)$ and $\xi_e\in\mathbb{R}$ a nontrivial edge dependent disorder. We show that in dimension $d=2$ there is a unique shift covariant disordered gradient Gibbs measure such that the annealed measure is ergodic and has zero tilt. This shows that the phase transitions known to occur for this class of potential do not persist to the disordered setting. The proof relies on the connection of the gradient Gibbs measures to a random conductance model with compact state space, to which the well known Aizenman-Wehr argument applies.Comment: 21 page

Renormalisation in discrete elasticity

This thesis deals with the statistical mechanics of lattice models. It has two main contributions. On the one hand we implement a general framework for a rigorous renormalisation group approach to gradient models. This approach relies on work by Bauerschmidt, Brydges, and Slade and extends earlier results for gradient interface models by Adams, Kotecký and Müller. On the other hand we use those results to analyse microscopic models for discrete elasticity at small positive temperature and in particular prove convexity properties of the free energy. The first Chapter is introductory and discusses the necessary mathematical background and the physical motivation for this thesis. Chapters 2 to 4 then contain a complete and almost self contained implementation of the renormalisation group approach for gradient models. Chapter 2 is concerned with a new construction of a finite range decomposition with improved regularity. Finite range decompositions are an important ingredient in the renormalisation group approach but also appear at various other places. The new finite range decomposition helps to avoid a loss of regularity and several technical problems that were present in the earlier applications of the renormalisation group technique to gradient models. In the third Chapter we analyse generalized gradient models and discrete models for elasticity and we state our main results: At low temperatures the surface tension is locally uniformly convex and the scaling limit is Gaussian. Moreover, we show that those statements can be reduced to a general statement about perturbations of massless Gaussian measures using suitable null Lagrangians. This is a first step towards a mathematical understanding of elastic behaviour of crystalline solids at positive temperatures starting from microscopic models. The fourth Chapter contains the renormalisation group analysis of gradient models. The main result is a bound for certain perturbations of Gaussian gradient measures that implies the results of the previous chapters. This generalizes earlier results for scalar nearest neighbour models to vector-valued finite range interactions. We also require a much weaker growth assumption for the perturbation. This is possible because we introduce a new solution to the large field problem based on an alternative construction of the weight functions using Gaussian calculus. The last Chapter has a slightly different focus. We investigate gradient interface models for a specific class of non-convex potentials for which phase transitions occur in dimension two. The analysis of these potentials is based on the relation to a random conductance model. We study properties of this random conductance model and in particular prove correlation inequalities and reprove the phase transition result relying on planar duality instead of reflection positivity

Design of a test environment for planning and interaction with virtual production processes

Rising complexity of systems combined with multi-disciplinary development and manufacturing processes necessitates new approaches of early validation of intermediate digital process and system prototypes. To develop and test these approaches, the modular digital cube test center was build. Usage of different Visualization Modules such as Powerwall, CAVE or Head Mounted Display allows immersive interaction with the prototypes. Combined with Haptic Interaction Modules from one axis assembly device to a hexapod simulator up to a full freedom kinematic portal and usage of different simulation modules of vehicle design, multi-kinematic, manufacturing and process-simulation allows early virtual prototypes validation in multiple use cases

Function Classes for Identifiable Nonlinear Independent Component Analysis

Unsupervised learning of latent variable models (LVMs) is widely used to represent data in machine learning. When such models reflect the ground truth factors and the mechanisms mapping them to observations, there is reason to expect that they allow generalization in downstream tasks. It is however well known that such identifiability guaranties are typically not achievable without putting constraints on the model class. This is notably the case for nonlinear Independent Component Analysis, in which the LVM maps statistically independent variables to observations via a deterministic nonlinear function. Several families of spurious solutions fitting perfectly the data, but that do not correspond to the ground truth factors can be constructed in generic settings. However, recent work suggests that constraining the function class of such models may promote identifiability. Specifically, function classes with constraints on their partial derivatives, gathered in the Jacobian matrix, have been proposed, such as orthogonal coordinate transformations (OCT), which impose orthogonality of the Jacobian columns. In the present work, we prove that a subclass of these transformations, conformal maps, is identifiable and provide novel theoretical results suggesting that OCTs have properties that prevent families of spurious solutions to spoil identifiability in a generic setting.Comment: 43 page

Multi armed bandits and quantum channel oracles

Multi armed bandits are one of the theoretical pillars of reinforcement learning. Recently, the investigation of quantum algorithms for multi armed bandit problems was started, and it was found that a quadratic speed-up is possible when the arms and the randomness of the rewards of the arms can be queried in superposition. Here we introduce further bandit models where we only have limited access to the randomness of the rewards, but we can still query the arms in superposition. We show that this impedes any speed-up of quantum algorithms.Comment: 44 page