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This doctoral thesis undertakes an in-depth exploration of limiting shape theorems across diverse mathematical structures, with a specific focus on subadditive processes within finitely generated groups exhibiting polynomial growth rates, as well as standard First-Passage Percolation (FPP) models applied to Random Geometric Graphs (RGGs). Employing a diverse range of techniques, including subadditive ergodic theorems and tailored modifications suited for polygonal paths within groups, the thesis examines the asymptotic shape under varying conditions. The investigation extends to subadditive cocycles characterized by at least and at most linear growth. Moreover, the study delves into moderate deviations for FPP models on RGGs, refining previous results with theorems that quantify its speed of convergence to the limiting shape, the fluctuation of the geodesics, and its spanning trees. Additionally, we apply the obtained results in a competition model to verify the positive probability of coexistence of two species competing for territory in a random geometric graph., Comment: PhD thesis, 146 pages, 13 figures. arXiv admin note: text overlap with arXiv:2312.07811

The transition dynamics of two-state systems with time-dependent energy levels, first considered by Landau, Zener, Majorana, and St\"uckelberg, is one of the basic models in quantum physics and has been used to describe various physical systems. We propose here a generalization of the Landau-Zener (LZ) problem characterized by distinct paths of the instantaneous eigenstates as the system evolves in time, while keeping the instantaneous eigenenergies exactly as in the standard LZ model. We show that these paths play an essential role in the transition probability $P$ between the two states, and can lead to superadiabatic transitions, i.e., to a substantial reduction of $P$. As an instructive extreme case, we identify an unconditionally adiabatic regime with $P=0$ no matter how fast the system evolves in time and regardless of the energy gap between the two levels at the anticrossing point. On the other hand, large $P$ occur even in the absence of any anticrossing point. These phenomena can be explained by observing the rotation of the instantaneous eigenvectors on the Bloch sphere. The superadiabatic LZ model can describe valley transition dynamics during charge and spin shuttling in semiconductor quantum dots., Comment: 6 pages, 3 figures

This study delves into the exploration of the limiting shape theorem for subadditive processes on finitely generated groups with polynomial growth, commonly referred to as virtually nilpotent groups. Investigating the algebraic structures underlying these processes, we present a generalized form of the asymptotic shape theorem within this framework. Extending subadditive ergodic theory in this context, we consider processes which exhibit both at most and at least linear random growth. We conclude with applications and illustrative examples., Comment: 36 pages, 1 figure; typos corrected, some lemmas revised

The valley splitting (VS) of a silicon quantum dot plays an important role for the performance and scalability of silicon spin qubits. In this work we investigate the VS of a SiGe/Si/SiGe heterostructure as a function of the size and location of the silicon quantum dot. We use the effective mass approach to describe a realistic system, which takes into account concentration fluctuations at the Si/SiGe interfaces and also the interface roughness. We predict that the size of the quantum dot is an important parameter for the enhancement of the VS and it can also induce a transition between the disorder-dominated to deterministic-enhanced regimes. Analyzing how the VS changes when we move the quantum dot in a specific direction, we obtain that the size of the quantum dot can be used to reduce the variability of the VS, which is relevant for charge/spin shuttling.

In this work we investigate the transport properties of non-relativistic quantum particles on incommensurate multilayered structures with the thicknesses $w_n$ of the layers following an extended Harper model given by $w_n = w_0 |\cos(\pi a n^{\nu})|$. For the normal incidence case, which means an one-dimensional system, we obtained that for a specific range of energy, it is possible to see a metallic-insulator transition with the exponent $\nu$. A metallic phase is supported for $\nu<1$. We also obtained that for the specific value $\nu=1$ there is an alternation between metallic and insulator phases as we change the disorder strength $w_0$. When we integrate out all incidence angles, which means a two-dimensional system, the metallic-insulator transition can be seen for much larger range of energy compared to the normal incidence case.

The performance and scalability of silicon spin qubits depend directly on the value of the conduction band valley splitting. In this work, we investigate the influence of electromagnetic fields and the interface width on the valley splitting of a quantum dot in a Si/SiGe heterostructure. We propose a new three-dimensional theoretical model within the effective mass theory for the calculation of the valley splitting in such heterostructures that takes into account the concentration fluctuation at the interfaces and the lateral confinement. With this model, we predict that the electric field is an important parameter for valley splitting engineering, since it can shift the probability distribution away from small valley splittings for some interface widths. We also obtain a critical softness of the interfaces in the heterostructure, above which the best option for spin qubits is to consider an interface as wide as possible.

Genetic and evolution algebras arise naturally from applied probability and stochastic processes. Gibbs measures describe interacting systems commonly studied in thermodynamics and statistical mechanics with applications in several fields. Here, we consider that the algebras are determined by configurations of finite spins on a countable set with their associated Gibbs distributions. The model preserves properties of the finite-dimensional Gibbs algebras found in the literature and extend their results. We introduce infertility in the genetic dynamics when the configurations differ macroscopically. It induces a decomposition of the algebra into a direct sum of fertile ideals with genetic realization. The proposed infinite-dimensional algebras are commutative, non-associative, with uncountable basis and zero divisors. The properties of Gibbs measures allow us to deal with the difficulties arising from the algebraic structure and obtain the results presented in this article., Comment: 26 pages, 4 figures; typos corrected, references added

We investigate the propagation of waves in one-dimensional systems with L\'evy-type disorder. We perform a complete analysis of non-relativistic and relativistic wave transmission submitted to potential barriers whose width, separation or both follow L\'evy distributions characterized by an exponent $0 < \alpha <1$. For the first two cases, where one of the parameters is fixed, non-relativistic and relativistic waves present anomalous localization, $\langle T \rangle \propto L^{-\alpha}$. However, for the latter case, in which both parameters follow a L\'evy distribution, non-relativistic and relativistic waves present a transition between anomalous and standard localization as the incidence energy increases relative to the barrier height. Moreover, we obtain the localization diagram delimiting anomalous and standard localization regimes, in terms of incidence angle and energy. Finally, we verify that transmission fluctuations, characterized by its standard deviation, are universal, independent of barrier architecture, wave equation type, incidence energy and angle, further extending earlier studies on electronic localization., Comment: Accepted for publication as a Regular Article in Physical Review E

We study the suppression of the conductance quantization in quantum spin Hall systems by a combined effect of electronic interactions and edge disorder, that is ubiquitous in exfoliated and CVD grown 2D materials. We show that the interplay between the electronic localized states due to edge defects and electron-electron interactions gives rise to local magnetic moments, that break time-reversal symmetry and the topological protection of the edge states in 2D topological systems. Our results suggest that edge disorder leads to small deviations of a perfect quantized conductance in short samples and to a strong conductance suppression in long ones. Our analysis is based on on the Kane-Mele model, an unrestricted Hubbard mean field Hamiltonian and on a self-consistent recursive Green's functions technique to calculate the transport quantities., Comment: 6 pages, 3 figures

Let a random geometric graph be defined in the supercritical regime for the existence of a unique infinite connected component in Euclidean space. Consider the first-passage percolation model with independent and identically distributed random variables on the random infinite connected component. We provide sufficient conditions for the existence of the asymptotic shape and we show that the shape is an Euclidean ball. We give some examples exhibiting the result for Bernoulli percolation and the Richardson model. For the Richardson model we further show that it converges weakly to a nonstandard branching process in the joint limit of large intensities and slow passage times., Comment: 23 pages, 3 figures. The references were upadted in the latest versions and typographical errors corrected

We define a stochastic reaction-diffusion process that describes a consensus formation in a non-sedentary population. The process is a diffusive version of the Majority Vote model, where the state update follows two stages: in the first stage, spins are allowed to hop to neighbor nodes with different probabilities for the respective spin orientation, and in the second stage, the spins in the same node can change its values according to the majority vote update rule. The model presents a consensus formation phase when concentration is greater than a threshold value, and a paramagnetic phase on the converse for equal diffusion probabilities, i.e., maintaining the inversion symmetry. The threshold vanishes for unequal diffusion probabilities, which means that the system has a consensus state for all values of population densities. The stationary collective opinion is dominated by the individuals that diffuse more., Comment: 13 pages, 6 figures

We theoretically study the thermoelectric transport properties of strained HgTe in the topological insulator phase. We developed a model for the system using a Dirac Hamiltonian including the effect of strain induced by the interface between HgTe and the CdTe substrate. The conductivity tensor was explored assuming the electrons are scattered by charge impurities, while the thermopower tensor was addressed using the Mott relation. Seebeck and Nernst responses exhibit remarkable enhancements in comparison with other two-dimensional Dirac materials, such as graphene, germanane, prosphorene and stanene. The intensity of these termoeletric responses, their dependencies with the external perpendicular magnetic field and temperature are also addressed.

We study the equilibrium and nonequilibrium electronic transport properties of multiprobe topological systems using a combination of the Landauer-B\"uttiker approach and nonequilibrium Green's functions techniques. We obtain general expressions for both nonequilibrium and equilibrium local electronic currents that, by suitable projections, allow one to compute charge, spin, valley, and orbital currents. We show that external magnetic fields give rise to equilibrium charge currents in mesoscopic system and study the latter in the quantum Hall regime. Likewise, a spin-orbit interaction leads to local equilibrium spin currents, that we analyze in the quantum spin Hall regime., Comment: 16 pages, 11 figures

The investigation of curved low-dimensional systems is a topic of great research interest. Such investigations include two-dimensional systems with cylindrical symmetry. In this work, we present a numerical study of the electronic transport properties of metallic nanotubes deviating from the cylindrical form either by having a bump or a depression, and under the influence of a magnetic field. Under these circumstances, it is found that the nanotube may be used as an energy high-pass filter for electrons. It is also shown that the device can be used to tune the angular momentum of transmitted electrons., Comment: The following article has been accepted by J. of Applied Physics

The search for new materials is a very intense task in many technological areas. In 2015, a new variant of graphene was proposed, the pentagraphene, which was followed by the propose of a pentagonal boron nitride structure called penta-BN$_2$. Based on these structures, we investigated the electronic, magnetic, and optical properties of penta-BN$_2$ nanoribbons (p-BNNRs) considering four different kinds of edges, carefully closing the valence shells with H atoms to prevent dangling bonds. To achieve this goal, we used first-principles calculations in a density functional theory framework. Our findings showed that the p-BNNRs have a rich magneto-electronic behavior, varying from semiconductor to half-metal. We obtained that they are ferrimagnetic, having an intrinsic magnetism, which allow potential applications in spintronic or spinwaves. From an optical absorption point of view, they mainly absorb at ultraviolet region of the spectrum, especially at UV-B region, which could indicate a potential application as a UV filter.

We introduce the conceptual formulation, design, fabrication, control and commercial translation with IoT connection of a hybrid-face social robot and validation of human emotional response to its affective interactions. The hybrid-face robot integrates a 3D printed faceplate and a digital display to simplify conveyance of complex facial movements while providing the impression of three-dimensional depth for natural interaction. We map the space of potential emotions of the robot to specific facial feature parameters and characterise the recognisability of the humanoid hybrid-face robot's archetypal facial expressions. We introduce pupil dilation as an additional degree of freedom for conveyance of emotive states. Human interaction experiments demonstrate the ability to effectively convey emotion from the hybrid-robot face to human observers by mapping their neurophysiological electroencephalography (EEG) response to perceived emotional information and through interviews. Results show main hybrid-face robotic expressions can be discriminated with recognition rates above 80% and invoke human emotive response similar to that of actual human faces as measured by the face-specific N170 event-related potentials in EEG. The hybrid-face robot concept has been modified, implemented, and released in the commercial IoT robotic platform Miko (My Companion), an affective robot with facial and conversational features currently in use for human-robot interaction in children by Emotix Inc. We demonstrate that human EEG responses to Miko emotions are comparative to neurophysiological responses for actual human facial recognition. Finally, interviews show above 90% expression recognition rates in our commercial robot. We conclude that simplified hybrid-face abstraction conveys emotions effectively and enhances human-robot interaction., Comment: This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible

In 2015, a new two dimensional (2D) carbon allotrope, called phagraphene, was theoretically proposed. Based on this structure, we propose here a new boron nitride structure called phaBN. It is composed by three types of rings: pentagons, hexagons and heptagons. We investigate the electronic, optical, vibrational and thermodynamic properties of phaBN using first-principles calculations in a density functional theory (DFT) framework. Our calculations revealed that the phaBN has an energy gap of 2.739 eV, which is almost half of the energy gap of the hexagonal boron nitride (h-BN), thus being a semiconductor material. By means of the optical, vibrational and thermodynamic properties, it was possible to observe the absorption interval, the stability of the structure and its formation process, respectively.

In this work, the divergence and curl operators are obtained using the coordinate free non rigid basis formulation of differential geometry. Although the authors have attempted to keep the presentation self-contained as much as possible, some previous exposure to the language of differential geometry may be helpful. In this sense the work is aimed to late undergraduate or beginners graduate students interested in mathematical physics. To illustrate the development, we graphically present the eleven coordinate systems in which the Laplace operator is separable. We detail the development of the basis and the connection for the cylindrical and paraboloidal coordinate systems. We also present in [1] codes both in Maxima and Maple for the spherical orthonormal basis, which serves as a working model for calculations in other situations of interest. Also in [1] the codes to obtain the coordinate surfaces are given.

In this paper, we investigate the spectrum of a class of weighted Laplacians on Cayley graphs and determine under what conditions the corresponding eigenspaces are generically irreducible. Specifically, we analyze the spectrum on left-invariant Cayley graphs endowed with an invariant metric, and we give some criteria for generically irreducible eigenspaces. Additionally, we introduce an operator that is comparable to the Laplacian and show that the same criterion holds., Comment: 18 pages, 2 figures; typos corrected

Over the past few years, different computer-aided diagnosis (CAD) systems have been proposed to tackle skin lesion analysis. Most of these systems work only for dermoscopy images since there is a strong lack of public clinical images archive available to design them. To fill this gap, we release a skin lesion benchmark composed of clinical images collected from smartphone devices and a set of patient clinical data containing up to 22 features. The dataset consists of 1,373 patients, 1,641 skin lesions, and 2,298 images for six different diagnostics: three skin diseases and three skin cancers. In total, 58.4% of the skin lesions are biopsy-proven, including 100% of the skin cancers. By releasing this benchmark, we aim to aid future research and the development of new tools to assist clinicians to detect skin cancer.

A transfer matrix approach is used to study the electronic transport in graphene superlattices with long-range correlated barrier spacements. By considering the low-energy electronic excitations as massless Dirac fermions, we compute by transmission spectra of graphene superlattices with potential barriers having spacements randomly distributed with long-range correlations governed by a power-law spectral density $S(k)\propto 1/k^{\alpha}$. We show that at large incidence angles, the correlations in the disorder distribution do not play a significant role in the electronic transmission. However, long-range correlations suppress the Anderson localization as normal incidence is approached and a band of transmitting modes sets up reminiscent of Klein tunneling.

In this work we investigate theoretically the influence of a Fermi velocity modulation in the electronic and transport properties of magnetic graphene superlattices. We solve the effective Dirac equation for graphene with a position dependent vector potential and Fermi velocity and use the transfer matrix method to obtain the transmission coefficient for the finite cases and the dispersion relation for a periodic superlattice. Our results reveals that the Fermi velocity modulation can control the resonance peaks of the transmittance and also works as a switch, turning on/off the transmission through the magnetic barriers. The results obtained here can be used for the fabrication of graphene-based electronic devices.

We introduce a new random walk with unbounded memory obtained as a mixture of the Elephant Random Walk and the Dynamic Random Walk which we call the Dynamic Elephant Random Walk (DERW). As a consequence of this mixture the distribution of the increments of the resulting random process is time dependent. We prove a strong law of large numbers for the DERW and, in a particular case, we provide an explicit expression for its speed. Finally, we give sufficient conditions for the central limit theorem and the law of the iterated logarithm to hold., Comment: We corrected a typo in the definition of the ERW

We study the quantum dynamics of ballistic electrons in rotating carbon nanotubes in the presence of a uniform magnetic field. When the field is parallel to the nanotube axis, the rotation-induced electric field brings about the spin-orbit interaction which, together with the kinetic, inertial, and Zeeman terms, compose the Schr\"odinger-Pauli Hamiltonian of the system. Full diagonalization of this Hamiltonian yields the eigenstates and eigenenergies leading to the calculation of the charge and spin currents. Our main result is the demonstration that, by suitably combining the applied magnetic field intensity and rotation speed, one can tune one of the currents to zero while keeping the other one finite, giving rise to a spin current generator.

In this work we investigate the effects of a Fermi velocity modulation in a valley filter in graphene created by a combination of a magnetic and electric barrier. With the effective Dirac equation of the system, we use the transfer matrix formalism to obtain the transmittance. We verify that the valley transport in graphene is very sensitive to a Fermi velocity modulation, which is able to choose which valley will be filtered with perfect filtering. Also, it is possible to use a Fermi velocity modulation to filter both valleys or to make the valley filter transparent. It reveals that the Fermi velocity is a powerful tool that can be used to tune a graphene valley filter, since it has a total control in its transport properties.

Previous works on deformed graphene predict the existence of valley-polarized states, however, optimal conditions for their detection remain challenging. We show that in the quantum Hall regime, edge-like states in strained regions can be isolated in energy within Landau gaps. We identify precise conditions for new conducting edges-like states to be valley polarized, with the flexibility of positioning them at chosen locations in the system. A map of local density of states as a function of energy and position reveals a unique braid pattern that serves as a fingerprint to identify valley polarization.

We study the frog model on Cayley graphs of groups with polynomial growth rate $D \geq 3$. The frog model is an interacting particle system in discrete time. We consider that the process begins with a particle at each vertex of the graph and only one of these particles is active when the process begins. Each activated particle performs a simple random walk in discrete time activating the inactive particles in the visited vertices. We prove that the activation time of particles grows at least linearly and we show that in the abelian case with any finite generator set the set of activated sites has a limiting shape., Comment: The original publication is available at www.esaim-ps.org

We study the minimal random walk introduced by Kumar, Harbola and Lindenberg. It is a random process on $\{0, 1, \ldots \}$ with unbounded memory which exhibits subdiffusive, diffusive and superdiffusive regimes. We prove the law of large numbers for the whole parameter set. Then we prove the central limit theorem and the law of the iterated logarithm for the minimal random walk under diffusive and marginally superdiffusive behaviors. More interestingly, we establish a result for the minimal random walk when it possesses the three regimes; we show the convergence of its rescaled version to a non-normal random variable., Comment: Published at https://iopscience.iop.org/article/10.1088/1742-5468/ab3343 in the Journal of Statistical Mechanics

Additively manufactured high strength aluminium alloy AA7075 was prepared using selective laser melting. High strength aluminium alloys prepared by selective laser melting have not been widely studied to date. The evolution of microstructure and hardness, with the attendant corrosion, were investigated. Additively manufactured AA7075 was investigated both in the as-produced condition and as a function of artificial ageing. The microstructure of specimens prepared was studied using electron microscopy. Production of AA7075 by selective laser melting generated a unique microstructure, which was altered by solutionising and further altered by artificial ageing - resulting in microstructures distinctive to that of wrought AA7075-T6. The electrochemical response of additively manufactured AA7075 was dependent on processing history, and unique to wrought AA7075-T6, whereby dissolution rates were generally lower for additively manufactured AA7075. Furthermore, immersion exposure testing followed by microscopy, indicated different corrosion morphology for additively manufactured AA7075, whereby resultant pit size was notably smaller, in contrast to wrought AA7075-T6., Comment: 37 pages, includes 4 Tables and 11 Figures

We investigate the propagation of electronic waves described by the Dirac equation subject to a L\'evy-type disorder distribution. Our numerical calculations, based on the transfer matrix method, in a system with a distribution of potential barriers show that it presents a phase transition from anomalous to standard to anomalous localization as the incidence energy increases. In contrast, electronic waves described by the Schr\"odinger equation do not present such transitions. Moreover, we obtain the phase diagram delimiting anomalous and standard localization regimes, in the form of an incidence angle versus incidence energy diagram, and argue that transitions can also be characterized by the behavior of the dispersion of the transmission. We attribute this transition to an abrupt reduction in the transmittance of the system when the incidence angle is higher than a critical value which induces a decrease in the transmission fluctuations., Comment: 6 pages, 6 figures, accepted for publication in Physical Review E