Eigenvalue order statistics for random Schrödinger operators with doubly-exponential tails

We consider random Schrödinger operators of the form Δ + ξ, where $Delta; is the lattice Laplacian on Zd and ξ is an i.i.d. random field, and study the extreme order statistics of the eigenvalues for this operator restricted to large but finite subsets of Zd. We show that for ξ with a doubly-exponential type of upper tail, the upper extreme order statistics of the eigenvalues falls into the Gumbel max-order class. The corresponding eigenfunctions are exponentially localized in regions where ξ takes large, and properly arranged, values. A new and self-contained argument is thus provided for Anderson localization at the spectral edge which permits a rather explicit description of the shape of the potential and the eigenfunctions. Our study serves as an input into the analysis of an associated parabolic Anderson problem

Faithful Squashed Entanglement

Squashed entanglement is a measure for the entanglement of bipartite quantum states. In this paper we present a lower bound for squashed entanglement in terms of a distance to the set of separable states. This implies that squashed entanglement is faithful, that is, strictly positive if and only if the state is entangled. We derive the bound on squashed entanglement from a bound on quantum conditional mutual information, which is used to define squashed entanglement and corresponds to the amount by which strong subadditivity of von Neumann entropy fails to be saturated. Our result therefore sheds light on the structure of states that almost satisfy strong subadditivity with equality. The proof is based on two recent results from quantum information theory: the operational interpretation of the quantum mutual information as the optimal rate for state redistribution and the interpretation of the regularised relative entropy of entanglement as an error exponent in hypothesis testing. The distance to the set of separable states is measured by the one-way LOCC norm, an operationally-motivated norm giving the optimal probability of distinguishing two bipartite quantum states, each shared by two parties, using any protocol formed by local quantum operations and one-directional classical communication between the parties. A similar result for the Frobenius or Euclidean norm follows immediately. The result has two applications in complexity theory. The first is a quasipolynomial-time algorithm solving the weak membership problem for the set of separable states in one-way LOCC or Euclidean norm. The second concerns quantum Merlin-Arthur games. Here we show that multiple provers are not more powerful than a single prover when the verifier is restricted to one-way LOCC operations thereby providing a new characterisation of the complexity class QMA.Comment: 24 pages, 1 figure, 1 table. Due to an error in the published version, claims have been weakened from the LOCC norm to the one-way LOCC nor

Eigenvalue order statistics for random Schrödinger operators with doubly-exponential tails

We consider random Schrödinger operators of the form Delta+zeta , where D is the lattice Laplacian on Zd and Delta is an i.i.d. random field, and study the extreme order statistics of the eigenvalues for this operator restricted to large but finite subsets of Zd. We show that for sigma with a doubly-exponential type of upper tail, the upper extreme order statistics of the eigenvalues falls into the Gumbel max-order class. The corresponding eigenfunctions are exponentially localized in regions where zeta takes large, and properly arranged, values. A new and self-contained argument is thus provided for Anderson localization at the spectral edge which permits a rather explicit description of the shape of the potential and the eigenfunctions. Our study serves as an input into the analysis of an associated parabolic Anderson problem

An Invitation to Generalized Minkowski Geometry

The present thesis contributes to the theory of generalized Minkowski spaces as a continuation of Minkowski geometry, i.e., the geometry of finite-dimensional normed spaces over the field of real numbers. In a generalized Minkowski space, distance and length measurement is provided by a gauge, whose definition mimics the definition of a norm but lacks the symmetry requirement. This seemingly minor change in the definition is deliberately chosen. On the one hand, many techniques from Minkowski spaces can be adapted to generalized Minkowski spaces because several phenomena in Minkowski geometry simply do not depend on the symmetry of distance measurement. On the other hand, the possible asymmetry of the distance measurement set up by gauges is nonetheless meaningful and interesting for applications, e.g., in location science. In this spirit, the presentation of this thesis is led mainly by minimization problems from convex optimization and location science which are appealing to convex geometers, too. In addition, we study metrically defined objects, which may receive a new interpretation when we measure distances asymmetrically. To this end, we use a combination of methods from convex analysis and convex geometry to relate the properties of these objects to the shape of the unit ball of the generalized Minkowski space under consideration

Optimal Inference with a Multidimensional Multiscale Statistic

We observe a stochastic process on [0,1]^ ( ≥ 1) satisfying ()=¹/²() + (), ∈ [0,1]^, where ≥ 1 is a given scale parameter (`sample size'), is the standard Brownian sheet on [0,1]^ and ∈ L₁([0,1]^) is the unknown function of interest. We propose a multivariate multiscale statistic in this setting and prove that the statistic attains a subexponential tail bound; this extends the work of 'Dumbgen and Spokoiny (2001)' who proposed the analogous statistic for =1. In the process, we generalize Theorem 6.1 of 'Dumbgen and Spokoiny (2001)' about stochastic processes with sub-Gaussian increments on a pseudometric space, which is of independent interest. We use the proposed multiscale statistic to construct optimal tests (in an asymptotic minimax sense) for testing = 0 versus (i) appropriate Hölder classes of functions, and (ii) alternatives of the form = __{_}$, where _ is an axis-aligned hyperrectangle in [0,1]^ and _ ∈ ℝ; _ and _ unknown. In Chapter 3 we use this proposed multiscale statistics to construct honest confidence bands for multivariate shape-restricted regression including monotone and convex functions

Optimal subgroup selection

In clinical trials and other applications, we often see regions of the feature space that appear to exhibit interesting behaviour, but it is unclear whether these observed phenomena are reflected at the population level. Focusing on a regression setting, we consider the subgroup selection challenge of identifying a region of the feature space on which the regression function exceeds a pre-determined threshold. We formulate the problem as one of constrained optimisation, where we seek a low-complexity, data-dependent selection set on which, with a guaranteed probability, the regression function is uniformly at least as large as the threshold; subject to this constraint, we would like the region to contain as much mass under the marginal feature distribution as possible. This leads to a natural notion of regret, and our main contribution is to determine the minimax optimal rate for this regret in both the sample size and the Type I error probability. The rate involves a delicate interplay between parameters that control the smoothness of the regression function, as well as exponents that quantify the extent to which the optimal selection set at the population level can be approximated by families of well-behaved subsets. Finally, we expand the scope of our previous results by illustrating how they may be generalised to a treatment and control setting, where interest lies in the heterogeneous treatment effect.Comment: 65 pages, 2 figures, to appear in the Annals of Statistic

Essays on Asymmetries in Contest

This thesis is concerned with the effects of asymmetries in ability and social preferences in contests and conflict networks. Standard models find that asymmetries monotonically decrease total and individual efforts. I demonstrate that this result does not necessarily hold when players are embedded in complex networks, have preferencesregardingthefairnessofthecontestortheoutcomesofothers, and when real subjects play these games in the lab. Chapter 1 formulates a network of bilateral contests in which locally unique equilibria always exist, and global uniqueness is possible. I find that an increase of one player’s ability can increase her effort and the effort of the entire network. If one player targets a specific opponent, other players follow. Chapter 2 imposes a budget constraint on this model. Most findings are robust to this modelling choice. This allows an investigation of topics like the effects of heterogeneity on team performance and the effect of asymmetries in the number of conflicts a player and her rivals are involved in. Chapter 3 documents that there exists no agreed way for implementing social preferences in contests. I derive four possible versions and critically assess their properties. When costs are considered, the magnitude of predicted overspreading and overbidding is reduced. Mild asymmetry can result in higher effort from the high ability player. In chapter 4, I present a pilot experiment in which social identities, with and without a hierarchy, are induced. We find that identities with such a hierarchy can trigger more aggressive play. To structure these findings, I suggest a foundational model of social preferences that relates them to social identity, where ‘close’ players are treated with altruism and ‘distant’ players are treated with spite

Square functions and radonifying operators

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Analysis in Banach Spaces Volume II : Probabilistic Methods and Operator Theory Preface

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