Recently, it has become increasingly clear that boundaries play a significant role in the understanding of the non-perturbative phase of the dynamics of strings. In this thesis we propose to study the effects of boundaries in non-critical string theory. We thus analyse boundary conformal field theories on random surfaces using the conformal gauge approach of David, Distler and Kawai. The crucial point is the choice of boundary conditions on the Liouville field. We discuss the Weyl anomaly cancellation for Polyakov's non-critical open bosonic string with Neumann, Dirichlet and free boundary conditions. Dirichlet boundary conditions on the Liouville field imply that the metric is discontinuous as the boundary is approached. We consider the semi-classical limit and argue how it singles out the free boundary conditions for the Liouville held. We define the open string susceptibility, the anomalous gravitational scaling dimensions and a new Yang-Mills Feynman mass critical exponent. Finally, we consider an application to the theory of non-critical dual membranes. We show that the strength of the leading stringy non-perturbative effects is of the order e(^-o(1/βst)), a result that mimics those found in critical string theory and in matrix models. We show how this restricts the space of consistent theories. We also identify non-critical one dimensional D-instantons as dynamical objects which exchange closed string states and calculate the order of their size. The extension to the minimal c ≤ 1 boundary conformal models is also briefly discussed.

The vortex fluctuations have proved to be responsible for the onset of dissipation in thin type-II superconducting Aims. There is also growing evidence that dissipation in high- temperature superconductors exhibits the same kind of two-dimensional (2D) behavior. However, a proper analysis of these materials requires a thorough understanding of the two-dimensional fluctuations. This thesis may be considered to consist of two parts. The first is concerned with two models that have often been used as models for 2D superconductors, the 2D Coulomb gas and the 2D XY model. The second part contains analyses related to high-temperature sup er conductivity. Through analysis of some renormalization equations for the Kosterlitz-Thouless (KT) transition, it is shown that the region governed by the KT critical behavior is very small and only applies at very low values for the flux-flow resistance. It is concluded that this critical behavior not is observable in superconductors, and, furthermore, that the only available method to test for 2D fluctuations at the onset of resistance, is through comparison with the 2D resistance scaling function. The critical temperature for the 2D XY model is determined by means of a finite- size scaling relation for the helicity modulus. The linearly screened potential in the XY model is written in terms of a correlation function. The analogy to the 2D Coulomb gas is found to be exact with a temperature-dependent bare interaction and a new expression for vorticity. It is also demonstrated that the Coulomb gas scaling concept may be applied to XY-type models. An analysis of resistance data for YBCO/PBCO superlattices in terms of the 2D resistance scaling function gives evidence for 2D behavior in the cases with large separation of the superconducting layers. In the superlattices with stronger interlayer coupling, the crossover to three-dimensional behavior is seen as a deviation from the scaling function as Tc is approached from above. The anisotropic three-dimensional (3D) XY model is examined as a model for high- temperature superconductors. It is shown that the density of vortices above Tc are closely equal in the anisotropic 3D model and the 2D model. This is taken as evidence that the 3D to 2D crossover found in the superlattices also is present in the anisotropic 3D XY model. / digitalisering@umu.se

L'objet principal de cette thèse est l'étude de plusieurs modèles de polynômes aléatoires. Il s'agit de comprendre le comportement macroscopique des racines de polynômes aléatoires dont le degré tend vers l'infini. Nous explorerons la connexion existant entre les racines de polynômes aléatoires et les gaz de Coulomb afin d'obtenir des principes de grandes déviations pour la mesure empiriques des racines. Nous revisitons l'article de Zeitouni et Zelditch qui établit un principe de grandes déviations pour un modèle général de polynômes aléatoires à coefficients gaussiens complexes. Nous étendons ce résultat au cas des coefficients gaussiens réels. Ensuite, nous démontrons que ces résultats restent valides pour une large classe de lois sur les coefficients, faisant des grandes déviations un phénomène universel pour ces modèles. De plus, nous démontrons tous les résultats précédents pour le modèle des polynômes de Weyl renormalisés. Nous nous intéressons aussi au comportement de la racine de plus grand module des polynômes de Kac. Celle-ci a un comportement non-universel et est en général une variable aléatoire à queues lourdes. Enfin, nous démontrons un principe de grandes déviations pour la mesure empirique des ensembles biorthogonaux. / The main topic of this thesis is the study of the roots of random polynomials from several models. We seek to understand the behavior of the roots as the degree of the polynomial tends to infinity. We explore the connexion between the roots of random polynomials and Coulomb gases to obtain large deviations principles for the empirical measures of the roots of random polynomials. We revisit the article of Zeitouni and Zelditch which establishes the large deviations for a rather general model of random polynomials with independent complex Gaussian coefficients. We extend this result to the case of real Gaussian coefficients. Then, we prove that those results are also valid for a wide class of distributions on the coefficients, which means that those large deviations principles are a universal property. We also prove all of those results for renormalized Weyl polynomials. study the largest root in modulus of Kac polynomials. We show that this random variable has a non-universal behavior and has heavy tails. Finally, we establish a large deviations principle for the empirical measures of biorthogonal ensembles.

Consideramos dois tópicos distintos relacionados a modelos clássicos da mecânica estatísticas de equilíbrio. O primeiro constitui-se na análise de equação parabólicas semi-lineares associadas à transformação de grupo de renormalização para o gás de Coulomb hierárquico bidimensional e o gás dipolos hierárquicos em dimensão d>1 após tomarmos um limite apropriado (limite L 1 do tamanho do bloco). O outro tópico estudado foi a construção de uma função majorante (, z) para a pressão termodinâmica de um gás formado por partículas interagentes com atividade z e temperatura -1, cuja interação entre dois corpos pode ser decomposta em escalas como um potencial estável. Somos capazes de demonstrar que o problema de valor inicial dado pela equação do gás de Coulomb está bem definido (existência, unicidade e dependência contínua das soluções) em um espaço funcional adequado e a solução converge assintoticamente para uma das infinitas contáveis soluções de equilíbrio. Quanto ao gás de dipolos, embora não tenhamos conseguido provar a existência e unicidade das soluções, garantimos que a única solução estacionária limitada inferiormente é a trivial nula, que é uma solução estável. Ao menos no caso dos modelos hierárquicos, os resultados obtidos permitem dar uma resposta definitiva à conjectura de Gallavotti e Nicolò sobre uma sequência infinita de transições de fase. A função majorante é construída como a solução de uma equação diferencial parcial quase-linear de primeira ordem. Através da do método das características relacionamos a solução (majorante) à função W de Lambert cuja expansão em série possui uma singularidade originada pelo corte que a função W possui no plano complexo. A descrição da função majorante como uma função W possui no plano complexo. A descrição da função majorante como uma função W permite uma melhora nas estimativas de raio de convergência para série de Mayer para pressão. / We have considered in this thesis two distinct topics related to classic models in equilibrium statistical mechanics. The first one is the analysis of semilinear parabolic partial differential equations given by a suitable limit (size of block L 1) in the renormalization group for the dipole gas in any dimension d>1. The other topic is the construction of a majorant function (, z) for the thermodynamic -1 whose potential admits a scale decomposition in terms of some stable potential. We are capable to demonstrate the well-posedness (existence, uniqueness and continuous dependence of solutions) for Coulomb gas equations and the global asymptotic convergence of the flow to one of its countably many equilibrium solutions. The dipole gas equations are technically more difficult and lack the results weve achieved in Coulomb gas but, despite its difficulties, we can establish the uniqueness of the trivial solution as a equilibrium ane and its stabilish. At least for hierarchical models, the established results give a definite answer to Gallovotti and Niclolòs conjecture of na infinite of phase transitions. The majorant function is constructed as the solution of a first order quase-linear partial differential equation. By means of the characteristics method we are able to relate its solution (the majorant) to Lamberts W-function whose series expansion possess a singularity given by W-function allows better estimates for Mayer series convergence.

Consideramos dois tópicos distintos relacionados a modelos clássicos da mecânica estatísticas de equilíbrio. O primeiro constitui-se na análise de equação parabólicas semi-lineares associadas à transformação de grupo de renormalização para o gás de Coulomb hierárquico bidimensional e o gás dipolos hierárquicos em dimensão d>1 após tomarmos um limite apropriado (limite L 1 do tamanho do bloco). O outro tópico estudado foi a construção de uma função majorante (, z) para a pressão termodinâmica de um gás formado por partículas interagentes com atividade z e temperatura -1, cuja interação entre dois corpos pode ser decomposta em escalas como um potencial estável. Somos capazes de demonstrar que o problema de valor inicial dado pela equação do gás de Coulomb está bem definido (existência, unicidade e dependência contínua das soluções) em um espaço funcional adequado e a solução converge assintoticamente para uma das infinitas contáveis soluções de equilíbrio. Quanto ao gás de dipolos, embora não tenhamos conseguido provar a existência e unicidade das soluções, garantimos que a única solução estacionária limitada inferiormente é a trivial nula, que é uma solução estável. Ao menos no caso dos modelos hierárquicos, os resultados obtidos permitem dar uma resposta definitiva à conjectura de Gallavotti e Nicolò sobre uma sequência infinita de transições de fase. A função majorante é construída como a solução de uma equação diferencial parcial quase-linear de primeira ordem. Através da do método das características relacionamos a solução (majorante) à função W de Lambert cuja expansão em série possui uma singularidade originada pelo corte que a função W possui no plano complexo. A descrição da função majorante como uma função W possui no plano complexo. A descrição da função majorante como uma função W permite uma melhora nas estimativas de raio de convergência para série de Mayer para pressão. / We have considered in this thesis two distinct topics related to classic models in equilibrium statistical mechanics. The first one is the analysis of semilinear parabolic partial differential equations given by a suitable limit (size of block L 1) in the renormalization group for the dipole gas in any dimension d>1. The other topic is the construction of a majorant function (, z) for the thermodynamic -1 whose potential admits a scale decomposition in terms of some stable potential. We are capable to demonstrate the well-posedness (existence, uniqueness and continuous dependence of solutions) for Coulomb gas equations and the global asymptotic convergence of the flow to one of its countably many equilibrium solutions. The dipole gas equations are technically more difficult and lack the results weve achieved in Coulomb gas but, despite its difficulties, we can establish the uniqueness of the trivial solution as a equilibrium ane and its stabilish. At least for hierarchical models, the established results give a definite answer to Gallovotti and Niclolòs conjecture of na infinite of phase transitions. The majorant function is constructed as the solution of a first order quase-linear partial differential equation. By means of the characteristics method we are able to relate its solution (the majorant) to Lamberts W-function whose series expansion possess a singularity given by W-function allows better estimates for Mayer series convergence.

A superconducting nanowire single-photon detector (SNSPD) is an emerging, and today commercially available technology, for photon-counting and quantum cryptography. Yet, the photon detection event is not fully understood and current modeling efforts require substantial computational resources which motivates studies of simpler models.  This thesis introduces a model for vortex dynamics in thin-layered superconductors, such as SNSPDs, using a simplified approach, which leads to a 2D Coulomb gas model where the vortices are modeled as electrostatic charges. The model is carefully constructed from the method of images to describe a wire with open boundary conditions and an applied supercurrent. Subsequently, equilibrium and non-equilibrium properties are sampled with the Metropolis-Hastings algorithm and further analyzed and discussed. The suggested model is shown to be effective and successfully reproduces expected SNSPD behavior; most importantly critical behavior and voltage pulses which are directly measured during detection events. In conclusion, a 2D Coulomb gas model can be a preferred alternative for modeling vortex dynamics in SNSPDs at a small computational cost, motivating further development and studies. / Supraledande nanotråd-enfotondetektorer (SNSPD:er) är en framväxande och idag kommersiellt tillgänglig teknologi som används för räknande av fotoner samt inom kvantkryptografi. Ändå är fotondetektionshändelsen inte helt förstådd och de nuvarande modelleringar kräver substantiell datorkraft vilket motiverar studier av enklare modeller. Det här examensarbetet introducerar en model för vortexdynamik i tunnskiktade supraledare såsom SNSPD:er genom ett förenklat tillvägagångssätt som leder till en 2D Coulombgas-modell där ett vortex modelleras som en elektrisk laddning. Modellen är noggrant konstruerad med öppna randvillkor och den så kallade frysta spegelbildsmetoden samt en pålagd superström. Då samlas mätvärden in på jämvikts- samt icke-jämviktsegenskaper hos systemet som vidare analyseras, jämförs och diskuteras. Den föreslagna modellen visas vara effektiv och reproducerar framgångsrikt förväntat beteende hos SNSPD:er; framför allt kritiskt beteende och spänningstoppar som direkt uppmäts i en fysisk detektionshändelse. Sammanfattningsvis, kan en 2D Coulombgas-modell vara ett föredraget alternativ för att modellera vortexdynamik hos SNSPD:er för en liten beräkningskostnad, vilket motiverar fler studier av detta.

Cette thèse s'intéresse aux phénomènes électrostatiques émergents dans les modèles magnétiques toroïdaux bi-dimensionnels à symétrie XY, fournissant ainsi un support pour de plus amples recherches dans le domaine de la transition de phase Berezinskii-Kosterlitz-Thouless (BKT).Dans de nombreux systèmes bi-dimensionnels, dont le modèle bi-dimensionnel XY du magnétisme, la transition BKT contrôle la dissociation thermique de paires de défauts topologiques liés. Le modèle XY est analogue au gaz de Coulomb bi-dimensionnel, à ceci près qu'il peut être simulé sans avoir à modéliser les interactions à longue distance du système Coulombien. Cette thèse élucide ce paradoxe en démontrant que l'approximation de Villain appliquée au modèle XY est strictement équivalente au modèle électrostatique de Maggs-Rossetto (MR) appliqué au système Coulombien bi-dimensionnel.Cette équivalence est utilisée pour sonder la transition BKT par l'application de l'algorithme MR au gaz de Coulomb bi-dimensionel. En simulant le système Coulombien, il est prouvé que les fluctuations dans l'organisation des charges autour du tore sont activées à la température de transition BKT. Ces fluctuations du champ électrique indiquent ainsi la phase de haute température de la transition.Il est ensuite montré que l'exposant critique effectif de la théorie de Bramwell-Holdsworth (BH) peut être mesuré dans les films d'hélium 4 superfluide, qui correspondent à des gaz de Coulomb effectifs dans la limite de systèmes de grandes tailles finies. / This thesis addresses the emergent electrostatics of two-dimensional, toroidal magnetic models that possess XY symmetry, providing a platform for novel investigations into the Berezinskii-Kosterlitz-Thouless (BKT) phase transition.The BKT transition drives the thermal dissociation of bound pairs of topological defects in many two-dimensional systems, including the two-dimensional XY model of magnetism. The XY model is closely analogous to the two-dimensional Coulomb gas, but can be simulated without computing the long-range interactions of the Coulombic system. This thesis elucidates this paradox by showing that Villain's approximation to the XY model is strictly equivalent to the Maggs-Rossetto (MR) electrostatic model when applied to the two-dimensional Coulomb gas.The mapping is used to probe the BKT transition through the application of the MR algorithm to the two-dimensional Coulomb gas. By simulating the Coulombic system, fluctuations in the winding of charges around the torus are shown to turn on at the BKT transition temperature. These topological-sector fluctuations in the electric field therefore signal the high-temperature phase of the transition.It is then shown that the effective critical exponent of Bramwell-Holdsworth (BH) theory can be measured in superfluid 4He films, which correspond to effective Coulomb gases in the limit of large but finite system size. With the Coulombic system taken as the base BKT system, it is inferred that BH theory is a general property of BKT systems.