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Doctoral Thesis
DOI
10.11606/T.45.2018.tde-05062018-155758
Document
Author
Full name
Antonio Marcos Batista do Nascimento
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2018
Supervisor
Committee
Fontes, Luiz Renato Goncalves (President)
Iambartsev, Anatoli
Picco, Jean-Georges Pierre
Proença, Rodrigo Bissacot
Vares, Maria Eulalia
Title in Portuguese
Tempo de chegada ao equilíbrio da dinâmica de Metropolis para o GREM
Keywords in Portuguese
Convergência ao equilíbrio
Desigualdade de Poincaré
Dinâmica de Metropolis
GREM
Lacuna espectral
Vidros de spins
Abstract in Portuguese
Neste trabalho consideramos um processo de Markov a tempo contínuo com espaço de estados finito em um meio aleatório, a saber, a dinâmica de Metropolis para o Modelo de Energia Aleatória Generalizado (GREM) com um número de níveis finito e discutimos o comportamento do seu tempo de chegada ao equilíbrio, o qual é dado pelo inverso da lacuna espectral de sua matriz de probabilidades de transição. No principal resultado desta tese provamos que o quociente entre o volume do sistema e o logaritmo do inverso da lacuna é quase sempre limitado, por cima, por uma função da temperatura, que também é a que descreve a energia livre do GREM sob o regime de temperaturas baixas. Como um estudo adicional, também é discutido um correspondente limitante inferior em um caso particular do GREM com 2 níveis.
Title in English
Reaching time to equilibrium of the Metropolis dynamics for the GREM
Keywords in English
Convergence to equilibrium
GREM
Metropolis dynamics
Poincaré inequality
Spectral gap
Spin glasses
Abstract in English
In this work we consider a finite state continuous-time Markov process in a random environment, namely, the Metropolis dynamics for the Generalized Random Energy Model (GREM) with a finite number of levels, and we discuss the behavior of its reaching time to equilibrium which is given by inverse of the spectral gap of its transition probability matrix. On the main result of this thesis, we prove the division between the system volume and the logarithm of the inverse of the gap is almost surely upper bounded by a function of the temperature that it is also the function that describe the free energy of the GREM at low temperature. As an additional study, it is also discuss the corresponding limiting lower in a particular case of the 2-level GREM.
 
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Publishing Date
2018-06-06
 
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