Nosso objeto de estudo são os sistemas de spins com interações de longo alcance; em particular, estamos interessados em sistemas cuja probabilidade invariante é o modelo de Ising em A^S, onde A = {-1, 1} é o espaço de spins e S = Z^d é o espaço de sítios. Apresentamos dois resultados originais que são consequências da aplicação de algoritmos de simulação perfeita e de acoplamento no contexto da construção deste tipo de sistemas e de suas respectivas probabilidades invariantes. / Our object of interest are spin systems with long-range interactions. As a special case, we are interested in systems whose invariant measure is the Ising model on A^S, where A = {-1, 1} is the space of spins and S = Z^d is the space of sites. We present two original results that are byproducts of the application of Perfect Simulation and Coupling algorithms in the context of the construction of these spin systems and their respective invariant measures.

Estudamos o comportamento de modelos para sistemas modulados magnéticos e estruturais. A primeira parte deste trabalho e dedicada ao modelo de Ising com interações competitivas numa rede de Bethe, no limite de coordenação infinita, num campo magnético. Focalizamos nossa atenção no comportamento das fases comensuráveis na presença de campo. Obtivemos vários diagramas T H utilizando algoritmos numéricos muito mais eficientes do que a simples iteração do mapeamento associado ao modelo. Na segunda parte estudamos o modelo de FrenkelKontorova com primeiro e segundo harmônicos no potencial externo. Encontramos e investigamos transições de segunda ordem no interior das fases comensuráveis de período ímpar. Essas transições, denominadas simétricaassimétricas, estão associadas à quebra da simetria por reflexão que ocorre para potenciais suficientemente fortes. / We studied the behavior of models for magnetically and structurally modulated systems. The first part of this work is dedicated to the study of the Ising model with competing interactions on a Bethe lattice, in the infinite coordination limit, in a magnetic field. We focused our attention on the behavior of the commensurate phases in the presence of a field. We obtained various T H phase diagrams using numerical methods far more efficient than simple iteration of the mapping associated to the model. In the second part we studied the FrenkelKontorova model with first and second harmonics in the external potential. We found and investigated the second order transitions within the commensurate phases of odd periodicity. These transitions, called symmetricasymmetric transitions, are related to the breaking of reflection symmetry which occurs at high potentials.

Made available in DSpace on 2015-04-22T22:07:28Z (GMT). No. of bitstreams: 1 VANUSA_ BEZERRA_ PACHECO.pdf: 1062027 bytes, checksum: 0625cb2a00e9f5171bb97d813a81f04d (MD5) Previous issue date: 2006-12-01 / Fundação de Amparo à Pesquisa do Estado do Amazonas / Neste trabalho investigamos o diagrama de fase do modelo de Ising de spin ½ aleatoriamente decorado nos planos de um filme fino de tamanho L. As interações nos planos simula a interação cobre-cobre (Cu-Cu) numa rede cúbica simples antiferromagnética, onde entre os vértices da rede coloca-se um spin decorador aleatoriamente distribuído, que simula o íon de oxigênio no plano de cobre-oxigênio (CuO2) de valor ½ e interagindo ferromagneticamente com os íons de cobre, provocando assim o fenômeno de frustração. Para este estudo, utilizamos a técnica do operador diferencial em aglomerado com um íon em conjunto com a aproximação do campo efetivo. Através dos diagramas de fase (formúla), onde (formúla) , que representa a relação das energias de interação ferromagnética da superfície com o bulk é possível notar um ponto multicrítico (formúla) que corresponde ao caso em que tanto a superfície quanto o bulk estão ordenados a um dado valor de concentração e valores para os parâmetros de frustrações (formúla) (parâmetro de frustração da superfície) e (formúla) (parâmetro de frustração do bulk). Para valores Δ < Δc, o sistema apresenta-se com bulk ordenado e a superfície desordenada, isto significa que a temperatura crítica do bulk ( b ) c T é maior que a temperatura crítica da superfície ( s ) c T , no entanto para Δ >Δc a superfície está ordenada e o bulk desordenado, isto é, . E para (formúla) verificamos que para determinados valores de concentração encontramos para qualquer valor de Δ os mesmos valores de temperaturas críticas.

We develop a cluster expansion in space-time for an infinite-dimensional system of interacting diffusions where the drift term of each diffusion depends on the whole past of the trajectory; these interacting diffusions arise when considering the Langevin dynamics of a ferromagnetic system submitted to a disordered external magnetic field.

No description available.

We provide a quantum algorithm for the exact evaluation of the Potts partition function for a certain class of restricted instances of graphs that correspond to irreducible cyclic codes. We use the same approach to demonstrate that quantum computers can provide an exponential speed up over the best classical algorithms for the exact evaluation of the weight enumerator polynomial for a family of classical cyclic codes. In addition to this we also provide an efficient quantum approximation algorithm for a function (signed-Euler generating function) closely related to the Ising partition function and demonstrate that this problem is BQP-complete. We accomplish the above for the Potts partition function by using a series of links between Gauss sums, classical coding theory, graph theory and the partition function. We exploit the fact that there exists an efficient approximation algorithm for Gauss sums and the fact that this problem is equivalent in complexity to evaluating discrete log. A theorem of McEliece allows one to turn the Gauss sum approximation into an exact evaluation of the Potts partition function. Stripping the physics from this result leaves one with the result for the weight enumerator polynomial. The result for the approximation of the signed-Euler generating function was accomplished by fashioning a new mapping between quantum circuits and graphs. The mapping provided us with a way of relating the cycle structure of graphs with quantum circuits. Using a slight variant of this mapping, we present the final result of this thesis which presents a way of testing families of quantum circuits for their classical simulatability. We thus provide an efficient way of deciding whether a quantum circuit provides any additional computational power over classical computation and this is achieved by exploiting the fact that planar instances of the Ising partition function (with no external magnetic field) can be efficiently classically computed.

We provide a quantum algorithm for the exact evaluation of the Potts partition function for a certain class of restricted instances of graphs that correspond to irreducible cyclic codes. We use the same approach to demonstrate that quantum computers can provide an exponential speed up over the best classical algorithms for the exact evaluation of the weight enumerator polynomial for a family of classical cyclic codes. In addition to this we also provide an efficient quantum approximation algorithm for a function (signed-Euler generating function) closely related to the Ising partition function and demonstrate that this problem is BQP-complete. We accomplish the above for the Potts partition function by using a series of links between Gauss sums, classical coding theory, graph theory and the partition function. We exploit the fact that there exists an efficient approximation algorithm for Gauss sums and the fact that this problem is equivalent in complexity to evaluating discrete log. A theorem of McEliece allows one to turn the Gauss sum approximation into an exact evaluation of the Potts partition function. Stripping the physics from this result leaves one with the result for the weight enumerator polynomial. The result for the approximation of the signed-Euler generating function was accomplished by fashioning a new mapping between quantum circuits and graphs. The mapping provided us with a way of relating the cycle structure of graphs with quantum circuits. Using a slight variant of this mapping, we present the final result of this thesis which presents a way of testing families of quantum circuits for their classical simulatability. We thus provide an efficient way of deciding whether a quantum circuit provides any additional computational power over classical computation and this is achieved by exploiting the fact that planar instances of the Ising partition function (with no external magnetic field) can be efficiently classically computed.

In this paper we introduce a model of default contagion in the financail market. The structure of the companies are represented by a Heavy Gravity Portfolio, where we assume there are N sectors in the market and in each sector i, there is one big trader and ni supply companies.The supply companies in each sector are directly inuenced by the bigtrader and the big traders are also pairwise interacting with each other.This development of the Ising model is called Heavy gravity portfolioand according to this, the relation between expectation and correlationof the default of companies are derived by means of simulations utilisingthe Gibbs sampler. Finally methods for maximum likelihood estimationand for a likelihood ratio test of the interaction parameter in the modelare derived.

No description available.

A majority of the data transmitted and processed today has an inherent structured high-dimensional nature, either because of the process of encoding using high-dimensional codebooks for providing a systematic structure, or dependency of the data on a large number of agents or variables. As a result, many problem setups associated with transmission and processing of data have a structured high-dimensional aspect to them. This dissertation takes a look at two such problems, namely, communication over networks using network coding, and learning the structure of graphical representations like Markov networks using observed data, from an information-theoretic perspective. Such an approach yields intuition about good coding architectures as well as the limitations imposed by the high-dimensional framework. Th e dissertation studies the problem of network coding for networks having multiple transmission sessions, i.e., multiple users communicating with each other at the same time. The connection between such networks and the information-theoretic interference channel is examined, and the concept of interference alignment, derived from interference channel literature, is coupled with linear network coding to develop novel coding schemes off ering good guarantees on achievable throughput. In particular, two setups are analyzed – the first where each user requires data from only one user (multiple unicasts), and the second where each user requires data from potentially multiple users (multiple multicasts). It is demonstrated that one can achieve a rate equalling a signi ficant fraction of the maximal rate for each transmission session, provided certain constraints on the network topology are satisfi ed. Th e dissertation also analyzes the problem of learning the structure of Markov networks from observed samples – the learning problem is interpreted as a channel coding problem and its achievability and converse aspects are examined. A rate-distortion theoretic approach is taken for the converse aspect, and information-theoretic lower bounds on the number of samples, required for any algorithm to learn the Markov graph up to a pre-speci fied edit distance, are derived for ensembles of discrete and Gaussian Markov networks based on degree-bounded graphs. The problem of accurately learning the structure of discrete Markov networks, based on power-law graphs generated from the con figuration model, is also studied. The eff ect of power-law exponent value on the hardness of the learning problem is deduced from the converse aspect – it is shown that discrete Markov networks on power-law graphs with smaller exponent values require more number of samples to ensure accurate recovery of their underlying graphs for any learning algorithm. For the achievability aspect, an effi cient learning algorithm is designed for accurately reconstructing the structure of Ising model based on power-law graphs from the con figuration model; it is demonstrated that optimal number of samples su ffices for recovering the exact graph under certain constraints on the Ising model potential values. / text