Este trabalho foca no estudo de duas redes complexas. A primeira é um modelo de Ising com campo aleatório. Este modelo segue uma distribuição de campo gaussiana e bimodal. Uma técnica de conectividade finita foi utilizada para resolvê-lo. Assim como um método de Monte Carlo foi aplicado para verificar os resultados. Há uma indicação em nossos resultados que para a distribuição gaussiana a transição de fase é sempre de segunda ordem. Para as distribuições bimodais há um ponto tricrítico, dependente do valor da conectividade . Abaixo de um certo mínimo de , só existe transição de segunda ordem. A segunda é uma rede neural atratora métrica. Mais precisamente, estudamos a capacidade deste modelo para armazenar os padrões estruturados. Em particular, os padrões escolhidos foram retirados de impressões digitais, que apresentam algumas características locais. Os resultados mostram que quanto menor a atividade de padrões de impressões digitais, maior a relação de carga e a qualidade de recuperação. Uma teoria, também foi desenvolvido como uma função de cinco parâmetros: a relação de carga, a conectividade, o grau de densidade da rede, a relação de aleatoriedade e a correlação do padrão espacial. / This work focus on the study of two complex networks. The first one is a random field Ising model. This model follows a gaussian and bimodal distribution, for the random field. A finite connectivity technique was utilized to solve it. As well as a Monte Carlo method was applied to verify our results. There is an indication in our results that for a gaussian distribution the phase transition is always second-order. For the bimodal distribution there is a tricritical point, tha depends on the value of the connectivity . Below a certain minimum , there is only a second-order transition. The second one is a metric attractor neural network. More precisely we study the ability of this model to learn structured patterns. In particular, the chosen patterns were taken from fingerprints, which present some local features. Our results show that the higher the load ratio and retrieval quality are the lower is the fingerprint patterns activity. A theoretical framework was also developed as a function of five parameters: the load ratio, the connectivity, the density degree of the network, the randomness ratio and the spatial pattern correlation.

Une donnée peut avoir diverses formes et peut provenir d'un large panel d'applications. Habituellement, une donnée possède beaucoup de bruit et peut être soumise aux effets du hasard. Les récents progrès en apprentissage automatique ont relancé les recherches théoriques sur les limites des différentes méthodes probabilistes de traitement du signal. Dans cette thèse, nous nous intéressons aux questions suivantes : quelle est la meilleure performance possible atteignable ? Et comment peut-elle être atteinte, i.e., quelle est la stratégie algorithmique optimale ?La réponse dépend de la forme des données. Les sujets traités dans cette thèse peuvent tous être représentés par des modèles graphiques. Les propriétés des données déterminent la structure intrinsèque du modèle graphique correspondant. Les structures considérées ici sont soit éparses, soit denses. Les questions précédentes peuvent être étudiées dans un cadre probabiliste, qui permet d'apporter des réponses typiques. Un tel cadre est naturel en physique statistique et crée une analogie formelle avec la physique des systèmes désordonnés. En retour, cela permet l'utilisation d'outils spécifiques à ce domaine et de résoudre des problèmes de satisfaction de contraintes et d'inférence statistique. La problématique de performance optimale est directement reliée à la structure des extrema de la fonction d'énergie libre macroscopique, tandis que les aspects algorithmiques proviennent eux de la minimisation de la fonction d'énergie libre microscopique (c'est-à-dire, dans la forme de Bethe).Cette thèse est divisée en quatre parties. Premièrement, nous aborderons par une approche de physique statistique le problème de la coloration de graphes aléatoires et mettrons en évidence un certain nombre de caractéristiques. Dans un second temps, nous calculerons une nouvelle limite supérieure de la taille de l'ensemble contagieux. Troisièmement, nous calculerons le diagramme de phase du modèle de Dawid et Skene dans la région dense en modélisant le problème par une factorisation matricielle de petit rang. Enfin, nous calculerons l'erreur optimale de Bayes pour une classe restreinte de l'estimation matricielle de rang élevé. / Datasets come in a variety of forms and from a broad range of different applications. Typically, the observed data is noisy or in some other way subject to randomness. The recent developments in machine learning have revived the need for exact theoretical limits of probabilistic methods that recover information from noisy data. In this thesis we are concerned with the following two questions: what is the asymptotically best achievable performance? And how can this performance be achieved, i.e., what is the optimal algorithmic strategy? The answer depends on the properties of the data. The problems in this thesis can all be represented as probabilistic graphical models. The generative process of the data determines the structure of the underlying graphical model. The structures considered here are either sparse random graphs or dense (fully connected) models. The above questions can be studied in a probabilistic framework, which leads to an average (or typical) case answer. Such a probabilistic formulation is natural to statistical physics and leads to a formal analogy with problems in disordered systems. In turn, this permits to harvest the methods developed in the study of disordered systems, to attack constraint satisfaction and statistical inference problems. The formal analogy can be exploited as follows. The optimal performance analysis is directly related to the structure of the extrema of the macroscopic free energy. The algorithmic aspects follow from the minimization of the microscopic free energy (that is, the Bethe free energy in this work) which is closely related to message passing algorithms. This thesis is divided into four contributions. First, a statistical physics investigation of the circular coloring problem is carried out that reveals several distinct features. Second, new rigorous upper bounds on the size of minimal contagious sets in random graphs, with bounded maximum degree, are obtained. Third, the phase diagram of the dense Dawid-Skene model is derived by mapping the problem onto low-rank matrix factorization. The associated approximate message passing algorithm is evaluated on real-world data. Finally, the Bayes optimal denoising mean square error is derived for a restricted class of extensive rank matrix estimation problems.

Este trabalho foca no estudo de duas redes complexas. A primeira é um modelo de Ising com campo aleatório. Este modelo segue uma distribuição de campo gaussiana e bimodal. Uma técnica de conectividade finita foi utilizada para resolvê-lo. Assim como um método de Monte Carlo foi aplicado para verificar os resultados. Há uma indicação em nossos resultados que para a distribuição gaussiana a transição de fase é sempre de segunda ordem. Para as distribuições bimodais há um ponto tricrítico, dependente do valor da conectividade . Abaixo de um certo mínimo de , só existe transição de segunda ordem. A segunda é uma rede neural atratora métrica. Mais precisamente, estudamos a capacidade deste modelo para armazenar os padrões estruturados. Em particular, os padrões escolhidos foram retirados de impressões digitais, que apresentam algumas características locais. Os resultados mostram que quanto menor a atividade de padrões de impressões digitais, maior a relação de carga e a qualidade de recuperação. Uma teoria, também foi desenvolvido como uma função de cinco parâmetros: a relação de carga, a conectividade, o grau de densidade da rede, a relação de aleatoriedade e a correlação do padrão espacial. / This work focus on the study of two complex networks. The first one is a random field Ising model. This model follows a gaussian and bimodal distribution, for the random field. A finite connectivity technique was utilized to solve it. As well as a Monte Carlo method was applied to verify our results. There is an indication in our results that for a gaussian distribution the phase transition is always second-order. For the bimodal distribution there is a tricritical point, tha depends on the value of the connectivity . Below a certain minimum , there is only a second-order transition. The second one is a metric attractor neural network. More precisely we study the ability of this model to learn structured patterns. In particular, the chosen patterns were taken from fingerprints, which present some local features. Our results show that the higher the load ratio and retrieval quality are the lower is the fingerprint patterns activity. A theoretical framework was also developed as a function of five parameters: the load ratio, the connectivity, the density degree of the network, the randomness ratio and the spatial pattern correlation.

Orientador: Eduardo Miranda / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin / Made available in DSpace on 2018-08-16T08:19:56Z (GMT). No. of bitstreams: 1 Andrade_EricdeCastroe_D.pdf: 5537554 bytes, checksum: 1391d5fcc710b5e471f0814a4a6d484f (MD5) Previous issue date: 2010 / Resumo: Sistemas eletrônicos fortemente correlacionados desordenados possuem dois mecanismos básicos para a localização eletrônica e a subsequente destruição do estado metálico: o de Mott (causado pela interação elétron-elétron) e o de Anderson (causado pela desordem). Nesta tese, estudamos como estes mecanismos competem dentro da fase metálica e também como afetam o comportamento crítico do sistema, empregando uma generalização para o caso desordenado do cenário de Brinkman-Rice para a transição de Mott. Investigamos os efeitos de desordem fraca e moderada sobre a transição metal-isolante de Mott a T = 0 em duas dimensões. Para desordem sucientemente baixa, a transição mantém sua característica do tipo Mott, na qual temos os pesos de quasipartícula Zi indo a zero na transição e uma forte blindagem da desordem na região crítica. Em contraste com o comportamento encontrado para d = 8 , no nosso caso as flutuações espaciais dos pesos de quasipartícula são fortemente amplificadas próximo à transição de Mott de tal forma que eles adquirem uma distribuição do tipo lei de potência P (Z) ~ Z a-1 ,com a --> 0 na transição. Tal comportamento altera completamente as características desta transição com relação ao caso limpo, e é um indício robusto da emergência de uma fase de Griffiths eletrônica precedendo a transição metal-isolante de Mott, com uma fenomenologia surpreendentemente similar àquela do "ponto fixo de desordem infinita" encontrada em magnetos quânticos. Uma consequência imediata dessas novas características introduzidas pela desordem é que estados eletrônicos próximos à superfície de Fermi tornam-se mais homogêneos na região crítica, ao passo que estados com maiores energias têm o comportamento oposto: eles apresentam uma grande inomogeneidade precisamente nas vizinhanças da transição de Mott. Sugerimos que uma desordem efetiva dependente da interação é uma característica comum a todos os sistemas de Mott desordenados. Estudamos também como os efeitos bem conhecidos das oscilações de longo alcance de Friedel são afetados por fortes correlações eletrônicas. Primeiramente, mostramos que sua amplitude e alcance são consideravelmente suprimidos em líquidos de Fermi fortemente renormalizados. Posteriormente, investigamos o papel dos espalhamentos elásticos e inelásticos na presença dessas oscilações. Em geral, nossos resultados analíticos mostram que um papel proeminente das oscilações de Friedel é relegado a sistemas fracamente interagentes. Abordamos, por m, os efeitos das interações sobre o isolante de Anderson em uma dimensão. Construímos a função de escala ß (g) e mostramos que a escala de "crossover" g *, que marca a transição entre o regime ôhmico e o localizado da condutância, é renormalizada pelas interações. Como consequência, embora não haja a emergência de estados verdadeiramente estendidos, o regime ôhmico de g estende-se agora por uma região consideravelmente maior do espaço de parâmetros. / Abstract: Disordered strongly correlated electronic systems have two basic routes towards localization underlying the destruction of the metallic state: the Mott route (driven by electron-electron interaction) and the Anderson route (driven by disorder). In this thesis, we study how these two mechanisms compete in the metallic phase, and also how they change the critical behavior of the system, within a generalization to the disordered case of the Brinkman-Rice scenario for the Mott transition. We investigate the effects of weak to moderate disorder on the Mott metal-insulator transition at T = 0 in two dimensions. For sufficiently weak disorder, the transition retains the Mott character, as signaled by the vanishing of the local quasiparticle weights Zi and strong disorder screening at criticality. In contrast to the behavior in d = 8, here the local spatial fluctuations of quasiparticle parameters are strongly enhanced in the critical regime, with a distribution function P(Z) ~ Z a-1 and a --> 0 at the transition. This behavior indicates the robust emergence of an electronic Griffiths phase preceding the MIT, in a fashion surprisingly reminiscent of the " Infinite Randomness Fixed Point" scenario for disordered quantum magnets. As an immediate consequence of these new features introduced by disorder, we have that the electronic states close to the Fermi energy become more spatially homogeneous in the critical region, whereas the higher energy states show the opposite behavior: they display enhanced spatial inhomogeneity precisely in the close vicinity to the Mott transition. We suggest that such energy-resolved disorder screening is a generic property of disordered Mott systems. We also study how well-known effects of the long-ranged Friedel oscillations are affected by strong electronic correlations. We first show that their range and amplitude are signifficantly suppressed in strongly renormalized Fermi liquids. We then investigate the interplay of elastic and inelastic scattering in the presence of these oscillations. In the singular case of two-dimensional systems, we show how the anomalous ballistic scattering rate is conned to a very restricted temperature range even for moderate correlations. In general, our analytical results indicate that a prominent role of Friedel oscillations is relegated to weakly interacting systems. Finally, we discuss the effects of correlations on the Anderson insulator in one dimension. We construct the scaling function ß(g) and we show that the crossover scaling g*, which marks the transition between the ohmic and the localized regimes of the conductance, is renormalized by the interactions. As a consequence, we show that, although truly extend states do not emerge, the ohmic regime covers now a considerably larger region in the parameter space. / Doutorado / Física da Matéria Condensada / Doutor em Ciências

Cette thèse présente plusieurs aspects de la croissance stochastique des interfaces, par lebiais de son modèle le plus étudié, l'équation de Kardar-Parisi-Zhang (KPZ). Bien qued'expression très simple, cette équation recèle une grande richesse phénoménologiqueet est l'objet d'une recherche intensive depuis des dizaines d'années. Cela a conduit àl'émergence d'une nouvelle classe d'universalité, contenant des modèles de croissanceparmi les plus courants, tels que le Eden model ou encore le Polynuclear Growth Model.L'équation KPZ est également reliée à des problèmes d'optimisation en présence dedésordre (le Polymère Dirigé), ou encore à la turbulence des uides (l'équation de Burger), renforçant son intérêt. Cependant, les limites de cette classe d'universalitésont encore mal comprises. L'objet de cette thèse est, après avoir présenté les progrèsles plus récents dans le domaine, de tester les limites de cette classe d'universalité. Lathèse s'articule en quatre parties :i) Dans un premier temps, nous présentons des outils théoriques qui permettent decaractériser finement l'évolution de l'interface. Ces outils montrent une grande flexibilité, que nous illustrons en considérant le cas d'une géométrie confinée (une interfacecroissant le long d'une paroi).ii) Nous nous penchons ensuite sur l'influence du désordre, et plus particulièrementl'importance des évènements extrêmes dans la mécanique de croissance. Les largesfluctuations du désordre déforment l'interface et conduisent à une modification notabledes exposants de scaling. Nous portons une attention particulière aux conséquencesd'un tel désordre sur les stratégies d'optimisation en milieu désordonné.iii) La présence de corrélations dans le désordre est d'un intérêt expérimentalimmédiat. Bien qu'elles ne modifient pas la classe d'universalité, elles influent grandement sur la vitesse de croissance moyenne de l'interface. Cette partie est dédiée àl'étude de cette vitesse moyenne, souvent négligée car délicate à définir, et à l'existenced'un optimum de croissance intimement lié à la compétition entre exploration et exploitation.iv) Enfin, nous considérons un exemple expérimental de croissance stochastique (quin'appartient toutefois pas à la classe KPZ) et développons un formalisme phénoménologiquepour modéliser la propagation d'une interface chimique dans un milieu poreux désordonné.Tout au long du manuscrit, les conséquences des phénomènes observées dans desdomaines variés, tels que les stratégies d'optimisation, la dynamique des populations,la turbulence ou la finance, sont détaillées. / This Thesis presents several aspects of the stochastic growth, through its most paradig-matic model, the Kardar-Parisi-Zhang equation (KPZ). Albeit very simple, this equa-tion shows a rich behaviour and has been extensively studied for decades. The existenceof a new universality class is now well established, containing numerous growth modelslike the Eden model or the Polynuclear Growth Model. The KPZ equation is closelyrelated to optimisation problems (the Directed Polymer) or turbulence of uids (theBurgers equation), a feature that underlines its importance. Nonetheless, the bound-aries of this universality class are still vague. The focus of this Thesis is to probe thoselimits through various modifications of the models. It is divided in four chapters:i) First, we present theoretical tools, borrowed from integrable systems, that allowto characterize in great details the evolution of the interface. Those tools exhibitconsiderable exibility due to the large corpus of work on integrable systems, and weillustrate it by tackling the case of confined geometry (growth close to a hard wall).ii) We investigate the inuence of the disorder distribution, and more specificallythe importance of large events, with heavy-tailed distributions. Those extreme eventsstretch the interface and notably modify the main scaling exponents. The consequenceson optimization strategies in disorder landscapes are emphasized.iii) The presence of correlations in the disorder is of natural experimental interest.Although they do not impact the KPZ class, they greatly inuence the average speed ofgrowth. The latter quantity is often overlooked because it is non-universal and ratherill-defined. Nonetheless, we show that a generic optimal average speed exists in presenceof time correlations, due to a competition between exploration and exploitation.iv) Finally, we consider a set of experiments about chemical front growth in porousmedium. While this growth process is not related to KPZ in an immediate way, wepresent different tools that effciently reproduce the observations.Along that work, the consequences of each Chapter in various domains, like opti-misation strategies, turbulence, population dynamics or finance, are detailed.

La physique statistique, développée à l'origine pour décrire les systèmes thermodynamiques, a joué pendant les dernières décennies un rôle central dans la modélisation d'un ensemble incroyablement vaste et hétérogène de différents phénomènes qui ont lieu par exemple dans des systèmes sociaux, économiques ou biologiques. Un champ d'applications possibles aussi vaste a été trouvé aussi pour les réseaux, comme une grande variété de systèmes peut être décrite en termes d'éléments interconnectés. Après une partie introductive sur les thèmes abordés ainsi que sur le rôle de la modélisation abstraite dans la science, dans ce manuscrit seront décrites les nouvelles perspectives auxquelles on peut arriver en approchant d'une façon physico-statistique trois problèmes d'intérêt dans la théorie des réseaux: comment une certaine quantité peut se répandre de façon optimale sur un graphique, comment explorer un réseau et comment le reconstruire à partir d'un jeu d'informations partielles. Quelques remarques finales sur l'importance que ces thèmes préserveront dans les années à venir conclut le travail. / Statistical physics, originally developed to describe thermodynamic systems, has been playing for the last decades a central role in modelling an incredibly large and heterogeneous set of different phenomena taking for instance place on social, economical or biological systems. Such a vast field of possible applications has been found also for networks, as a huge variety of systems can be described in terms of interconnected elements. After an introductory part introducing these themes as well as the role of abstract modelling in science, in this dissertation it will be discussed how a statistical physics approach can lead to new insights as regards three problems of interest in network theory: how some quantity can be optimally spread on a graph, how to explore it and how to reconstruct it from partial information. Some final remarks on the importance such themes will likely preserve in the coming years conclude the work.

Ce mémoire présente nos travaux sur les simulations numériques des mécanismes de dissipation mécanique interne (DMI) dans le a-Si. Ce travail s’inscrit dans le contexte des détecteurs d’ondes gravitationnelles, où les excitations à basses énergies dans les matériaux des miroirs constituent la principale source de bruit. On introduit le cadre théorique dans lequel le mémoire s’inscrit, soit le théorème de fluctuation-dissipation et les théories de l’état de transition et systèmes à deux niveaux, et on fait un court résumé de l’état des connaissances expérimentales sur le sujet. On présente ensuite les méthodes numériques : les méthodes d’exploration de l’énergie potentielle, les potentiels interatomiques et les méthodes de préparation des configurations atomiques, de même qu’une revue des travaux théoriques sur la DMI. Les résultats principaux du projet de maîtrise, incluant l’analyse des systèmes à deux niveaux dans le a-Si et le calcul de la DMI, sont présentés au troisième chapitre, sous la forme d’un article de revue. On termine par détailler nos travaux sur la DMI en employant la spectroscopie mécanique. / This master’s thesis presents our work on numerical simulations of internal mechanical dissipation (IMD) mechanisms in a-Si. This work is done in the context of gravitational wave detectors, where low-energy excitations in the materials of the mirrors are the main source of noise. We introduce the theoretical framework of the project, starting with the fluctuationdissipation theorem, and following with the transition state theory and the two-level systems (TLS) theory. A short review of experimental work on the subject is also presented. This is followed by a presentation of the numerical methods: potential energy landscape exploration techniques, interatomic potentials and atomic configurations preparation methods, as well as a review of numerical studies of the IMD. The main results of the thesis, an analysis of the two-level systems in a-Si and calculations of the IMD, are presented in chapter 3, in the form of a journal article. We finish by detailing our work on IMD using mechanical spectroscopy.

The results reported in this thesis contribute to the understanding of disordered systems, to mesoscopic physics on the one hand, and to the physics of spin glasses on the other hand. The first part of this thesis studies numerically coherent electronic transport in a non magnetic metal accurately doped with frozen magnetic impurities (a low temperature spin glass). Thanks to a recursive code that calculates the two terminal conductance of the system, we study in detail the metallic regime of conduction (large conductance) as well as the insulating regime (small conductance). In both regimes, we highlight a universal behavior of the system. Moreover, a study of correlations between the conductance of different spin configurations of impurities allows us to link these correlations with correlations between spin configurations. This study opens the route for the first experimental determination of the overlap via transport measurements. A second part of this thesis deals with the study of the mean field Sherrington-Kirkpatrick model, which describes the low temperature phase of an Ising spin glass. We are interested here in the generalization of this model to quantum spins (i.e including the possibility to flip by quantum tunneling) of this classical model that was well studied during the past thirty years. We deduce analytically motion equations at the semi-classical level, for which the influence of quantum tunneling is weak, and we compare them with the classical case. We finally solve numerically these equations using a pseudo-spectral method.

Condensed matter systems undergoing phase transitions rarely allow exact solutions. The presence of disorder renders the situation  even worse but collective Monte Carlo methods and parallel algorithms allow numerical descriptions. This thesis considers classical phase transitions in disordered spin systems in general and in effective models of superfluids with disorder and novel interactions in particular. Quantum phase transitions are considered via a quantum to classical mapping. Central questions are if the presence of defects changes universal properties and what qualitative implications follow for experiments. Common to the cases considered is that the disorder maps out correlated structures. All results are obtained using large-scale Monte Carlo simulations of effective models capturing the relevant degrees of freedom at the transition. Considering a model system for superflow aided by a defect network, we find that the onset properties are significantly altered compared to the $\lambda$-transition in $^{4}$He. This has qualitative implications on expected experimental signatures in a defect supersolid scenario. For the Bose glass to superfluid quantum phase transition in 2D we determine the quantum correlation time by an anisotropic finite size scaling approach. Without a priori assumptions on critical parameters, we find the critical exponent $z=1.8 \pm 0.05$ contradicting the long standing result $z=d$. Using a 3D effective model for multi-band type-1.5 superconductors we find that these systems possibly feature a strong first order vortex-driven phase transition. Despite its short-range nature details of the interaction are shown to play an important role. Phase transitions in disordered spin models exposed to correlated defect structures obtained via rapid quenches of critical loop and spin models are investigated. On long length scales the correlations are shown to decay algebraically. The decay exponents are expressed through known critical exponents of the disorder generating models. For cases where the disorder correlations imply the existence of a new long-range-disorder fixed point we determine the critical exponents of the disordered systems via finite size scaling methods of Monte Carlo data and find good agreement with theoretical expectations. / <p>QC 20150306</p>

Considere $N$ pontos distribuídos de forma aleatória e uniforme num hipercubo $d$-dimensional. Cada ponto representa um sítio num meio desordenado. Um caminhante explora este meio saltando para os sítios mais próximos, que não tenham sido visitados nos últimos $\\mu$ (memoria) passos, inclusive o próprio sítio. A trajetória do caminhante é composta de uma parte transiente e de uma parte periódica (ciclos). Neste caso, o viajante pode ou não explorar todos espaço disponível. A partir de uma memória crítica, ocorre uma transição entre os regimes de exploração localizado e estendido. Para sistemas unidimensionais, essa transição ocorre na memória crítica $\\mu_1=\\log_2{N}$. A regra determinista pode ser suavizada, a fim de considerar situações mais realistas, com a inclusão do parâmetro estocástico $T$ (temperatura). Agora, os movimentos do caminhante são definidos por uma função densidade de probabilidade (PDF) que é parametrizada por $T$ e por uma função custo, que cresce à medida que a distância entre os sítios cresce. A PDF é escolhida de forma a favorecer saltos para sítios mais próximos. Com o aumento da temperatura, o caminhante pode sair de ciclos e estender sua exploração. Aqui, nós apresentamos os estudos analíticos e numéricos sobre a influência da temperatura e da memória crítica na exploração de um meio desordenado unidimensional. / Consider $N$ sites randomly and uniformly distributed in a $d$-dimensional hypercube. A walker explores this disordered medium going to the nearest site, which has not been visited in the last $\\mu$ (memory) steps. The walker trajectory is composed of a transient part and a periodic part (cycles). In this case, travelers can or cannot explore all available space, given rise to a crossover at critical memory, for one-dimensional systems $\\mu_1=\\log_2{N}$, between localized and extended regimes. % as function of $\\mu$. The deterministic rule can be softened to consider more realistic situations with the inclusion of a stochastic parameter $T$ (temperature). In this case, the walker movement is defined by a probability density function (PDF) that is parameterized by $T$ and a cost function, which increases as the distance among sites increases. The PDF is chosen to favor hops to nearest sites. As the temperature increases, the walker can escape from cycles and extend the exploration. Here we report the analytical and numerical studies of the influence of the temperature and the critical memory in the exploration of a one-dimensional disordered system.