Spin Glasses, Boolean Satisfiability, and Survey Propagation

In recent years statistical physics and computational complexity have found mutually interesting subjects of research. The theory of spin glasses from statistical physics has been successfully applied to the boolean satisfiability problem, which is the canonical topic of computational complexity. The study of spin glasses originated from experimental studies of the magnetic properties of impure metallic alloys, but soon the study of the theoretical models outshone the interest in the experimental systems. The model studied in this thesis is that of Ising spins with random interactions. In this thesis we discuss two analytical derivations on spin glasses: the famous replica trick on the Sherrington-Kirkpatrick model and the cavity method on a Bethe lattice spin glass. Computational complexity theory is a branch of theoretical computer science that studies how the running time of algorithms scales with the size of the input. Two important classes of algorithms or problems are P and NP, or colloquially easy and hard problems. The first problem to be proven to belong to the class of NP-complete problems is that of boolean satisfiability, i.e., the study of whether there is an assignment of variables for a random boolean formula so that the formula is satisfied. The boolean satisfiability problem can be tackled with spin glass theory; the cavity method can be applied to it. Boolean satisfiability exhibits a phase transition. As one increases the ratio of constraints to variables the probability of a random formula being satisfiable drops from unity to zero. This transition of random formulas from satisfiable to unsatisfiable is continuous for small formulas. It grows sharper with increasing problem size and becomes discrete at the limit of an infinite number of variables. The cavity method gives a value for the location of the phase transition that is in agreement with the numerical value. The cavity method is an analytical tool for studying average values over a distribution, but it introduces so called surveys that can also be calculated numerically for a single instance. These surveys inspire the survey propagation algorithm that is implemented as a numerical program to efficiently solve large instances of random boolean satisfiability problems. In this thesis I present a parallel version of survey propagation that achieves a speedup by a factor of 3 with 4 processors. With the improved version we are able to gain further knowledge on the detailed workings of survey propagation. It is found, firstly, that the number of iterations needed for one convergence of survey propagation depends on the number of variables, seemingly as ln(N). Secondly, it is found that the constraint to variable ratio for which survey propagation succeeds is dependent on the number of variables

Gravity Waves as a Probe of Hubble Expansion Rate During An Electroweak Scale Phase Transition

Just as big bang nucleosynthesis allows us to probe the expansion rate when the temperature of the universe was around 1 MeV, the measurement of gravity waves from electroweak scale first order phase transitions may allow us to probe the expansion rate when the temperature of the universe was at the electroweak scale. We compute the simple transformation rule for the gravity wave spectrum under the scaling transformation of the Hubble expansion rate. We then apply this directly to the scenario of quintessence kination domination and show how gravity wave spectra would shift relative to LISA and BBO projected sensitivities.Comment: 28 pages, 2 figures

The economic impact of military expenditures

The author addresses three questions about military spending in developing countries: What are the levels of (and trends in) military spending as a percentage of gross national product? What impact does peacetime military spending have on growth, government spending on social welfare and infrastructure, and other key economic variables? What major factors influence the level of military spending? The author finds that military spending as a share of GNP generally fell in the 1980s, even in the Middle East and North Africa. The mean level of military expenditure as a share of GNP (MES) was 3.9 percent, well below the peak of 5.3 percent in 1976. In 1989, MES averaged only 2.7 percent in Latin America and 2.0 percent in sub-Saharan Africa - the two regions with the most severe economic problems. He finds no evidence of a negative relationship between military spending as a share of GNP and the peacetime growth rate of developing countries - except where military spending is high. He finds that higher shares of MES are not associated with lower shares of government spending on education, health, and infrastructure. As MES increases, government spending as a share of GNP increases, which allows the level of spending on health, education, and infrastructure to be maintained. The author finds some evidence that increased military spending in the developing countries has a weak negative impact on investment and the balance of trade. He finds no evidence of a statistically significant relationship between military spending and inflation. The most important determinant of peacetime military spending is the spending level of neighboring countries - in other words, the potential external threat. Regional conciliation and disarmament may be an important step toward reduced military spending.Achieving Shared Growth,Economic Growth,Peace&Peacekeeping,Health Monitoring&Evaluation,Economic Theory&Research

Equivalence Principle in Chameleon Models

Most theories that predict time and/or space variation of fundamental constants also predict violations of the Weak Equivalence Principle. In 2004 Khoury and Weltman proposed the so called chameleon field arguing that it could help avoiding experimental bounds on the WEP while having a non-trivial cosmological impact. In this paper we revisit the extent to which these expectations continue to hold as we enter the regime of high precision tests. The basis of the study is the development of a new method for computing the force between two massive bodies induced by the chameleon field which takes into account the influence on the field by both, the large and the test bodies. We confirm that in the thin shell regime the force does depend non-trivially on the test body\' s composition, even when the chameleon coupling constants are universal. We also propose a simple criterion based on energy minimization, that we use to determine which of the approximations used in computing the scalar field in a two body problem is better in each specific regime. As an application of our analysis we then compare the resulting differential acceleration of two test bodies with the corresponding bounds obtained from E\"otv\"os type experiments. We consider two setups: 1) an Earth based experiment where the test bodies are made of Be and Al; 2) the Lunar Laser Ranging experiment. We find that for some choices of the free parameters of the chameleon model the predictions of the E\"otv\"os parameter are larger than some of the previous estimates. As a consequence, we put new constrains on these free parameters. An important result of our analysis is that our approach leads to new constraints on the parameter space of the chameleon models.Comment: 42 pages, 15 figures Accepted for publication in PR

Composite Majorana Fermion Wavefunctions in Nanowires

We consider Majorana fermions (MFs) in quasi-one-dimensional nanowire systems containing normal and superconducting sections where the topological phase based on Rashba spin orbit interaction can be tuned by magnetic fields. We derive explicit analytic solutions of the MF wavefunction in the weak and strong spin orbit interaction regimes. We find that the wavefunction for one single MF is a composite object formed by superpositions of different MF wavefunctions which have nearly disjoint supports in momentum space. These contributions are coming from the extrema of the spectrum, one centered around zero momentum and the other around the two Fermi points. As a result, the various MF wavefunctions have different localization lengths in real space and interference among them leads to pronounced oscillations of the MF probability density. For a transparent normal-superconducting junction we find that in the topological phase the MF leaks out from the superconducting into the normal section of the wire and is delocalized over the entire normal section, in agreement with recent numerical results by Chevallier et al. (arXiv:1203.2643)

Local tests of global entanglement and a counterexample to the generalized area law

We introduce a technique for applying quantum expanders in a distributed fashion, and use it to solve two basic questions: testing whether a bipartite quantum state shared by two parties is the maximally entangled state and disproving a generalized area law. In the process these two questions which appear completely unrelated turn out to be two sides of the same coin. Strikingly in both cases a constant amount of resources are used to verify a global property.Comment: 21 pages, to appear FOCS 201