Interpolating between random walk and rotor walk

We introduce a family of stochastic processes on the integers, depending on a parameter $p \in [0,1]$ and interpolating between the deterministic rotor walk (p=0) and the simple random walk (p=1/2). This p-rotor walk is not a Markov chain but it has a local Markov property: for each $x \in \mathbb{Z}$ the sequence of successive exits from $x$ is a Markov chain. The main result of this paper identifies the scaling limit of the p-rotor walk with two-sided i.i.d. initial rotors. The limiting process takes the form $\sqrt{\frac{1-p}{p}} X(t)$, where $X$ is a doubly perturbed Brownian motion, that is, it satisfies the implicit equation \begin{equation} X(t) = \mathcal{B}(t) + a \sup_{s\leq t} X(s) + b \inf_{s\leq t} X(s) \end{equation} for all $t \in [0,\infty)$. Here $\mathcal{B}(t)$ is a standard Brownian motion and $a,b<1$ are constants depending on the marginals of the initial rotors on $\mathbb{N}$ and $-\mathbb{N}$ respectively. Chaumont and Doney [CD99] have shown that the above equation has a pathwise unique solution $X(t)$, and that the solution is almost surely continuous and adapted to the natural filtration of the Brownian motion. Moreover, $\limsup X(t) = +\infty$ and $\liminf X(t) = -\infty$ [CDH00]. This last result, together with the main result of this paper, implies that the p-rotor walk is recurrent for any two-sided i.i.d. initial rotors and any $0<p<1$.Comment: 22 pages, 2 figures; Remark about the connection between our model and excited random walks with Markovian cookie stacks added. References adde

Random rotor walks and i.i.d. sandpiles on Sierpinski graphs

We prove that, on the infinite Sierpinski gasket graph SG, rotor walk with random initial configuration of rotors is recurrent. We also give a necessary condition for an i.i.d. sandpile to stabilize. In particular, we prove that an i.i.d. sandpile with expected number of chips per site greater or equal to three does not stabilize almost surely. Furthermore, the proof also applies to divisible sandpiles and shows that divisible sandpile at critical density one does not stabilize almost surely on SG.Comment: 13 pages, 3 figures; referee's suggestions implemented; to appear in Statistics and Probability Letters (2024

Monte Carlo evaluation of the equilibrium isotope effects using the Takahashi-Imada factorization of the Feynman path integral

The Feynman path integral approach for computing equilibrium isotope effects and isotope fractionation corrects the approximations made in standard methods, although at significantly increased computational cost. We describe an accelerated path integral approach based on three ingredients: the fourth- order Takahashi-Imada factorization of the path integral, thermodynamic integration with respect to mass, and centroid virial estimators for relevant free energy derivatives. While the frst ingredient speeds up convergence to the quantum limit, the second and third improve statistical convergence. The combined method is applied to compute the equilibrium constants for isotope exchange reactions H2+D=H+HD and H2+D2=2HD

Spectral statistics across the many-body localization transition

The many-body localization transition (MBLT) between ergodic and many-body localized phase in disordered interacting systems is a subject of much recent interest. Statistics of eigenenergies is known to be a powerful probe of crossovers between ergodic and integrable systems in simpler examples of quantum chaos. We consider the evolution of the spectral statistics across the MBLT, starting with mapping to a Brownian motion process that analytically relates the spectral properties to the statistics of matrix elements. We demonstrate that the flow from Wigner-Dyson to Poisson statistics is a two-stage process. First, fractal enhancement of matrix elements upon approaching the MBLT from the metallic side produces an effective power-law interaction between energy levels, and leads to a plasma model for level statistics. At the second stage, the gas of eigenvalues has local interaction and level statistics belongs to a semi-Poisson universality class. We verify our findings numerically on the XXZ spin chain. We provide a microscopic understanding of the level statistics across the MBLT and discuss implications for the transition that are strong constraints on possible theories.Comment: 5 pages, 3 figure

Modernisations and digitalizations of fast timing techniques for γ - γ and e− - γ timing and lifetime measurements in middle-heavy rare earths nuclei

This doctoral thesis presents the modernisation and digitalization of two different experimental techniques and the results of different nuclear structure experiments in: The Orange conversion electron spectrometer setup was revised, rebuild in many hardware units and the usability has been restored by extensive software development of a control and acquisition software. Two different measurement modi are supported: a scan modes, which allows for conversion electron spectroscopy of an energy range during γ-source and in-beam measurements and an continuous mode, where one conversion electron energy can be selected by the strenghts of the magnetic field generated by the Orange spectrometer. Additionally, the implementation and validation of the digital fast timing technique with the recently commissioned fast-sampling digitizers Caen V1730 and additionally V1751 has been carried out. The digitizers have been installed and thoroughly been investigated and characterized in terms of time resolution and time walk and the settings of the integrated digital constant fraction discriminators have been optimized for digital fast-timing experiments. Both developments have successfully been commissioned in e− – γ and γ – γ fast-timing lifetime measurements in nuclei belonging to the neutron midshell N = 104, A ≈ 180 region. The lifetime results of four tungsten and osmium isotopes are discussed within the context of the deformation and transitional phenomena in the neutron midshell N = 104 region. All investigated nuclei lie at the edge of the strongly deformed region whose center is represented by the midshell-midshell nucleus 170 Dy. For 176,178,180 W, fast-timing methods using the newly revised and commissioned Orange conversion electron spectrometer and the Cologne HORUS spectrometer were employed. Different fusion evaporation reactions were utilized to populate excited states in these tungsten isotopes, allowing + + + for the measurement of lifetimes for the 2+ 1 , 41 , 61 , and 81 yrast states, with an additional 176 W. The extracted quadrupole deformation parameters, measurement of the 10+ 1 state in reduced transition probabilities, and B4/2 ratios were compared with Interacting Boson Model 1 calculations, confirming the deformed prolate rotor nature of the investigated tungsten isotopes. + + In the case of 182 Os the low-lying yrast states 2+ 1 , 41 , and 61 were studied using the newly implemented and commissioned digital fast-timing technique of the IKP. For the first time, + + lifetimes were determined for the 4+ 1 and 61 states, while the lifetime of the 21 state was re-evaluated to remove inconsistencies in the literature. The obtained lifetimes and extracted B(E2) values were analyzed in terms of collective signatures and transitional phenomena. The + + + calculated B(E2; 4+ 1 → 21 )/B(E2;21 → 01 ) ratio of 1.39(7) supported the interpretation of 182 Os as a rigid rotor, and its significance was discussed alongside neighboring isotopes and isotones within the framework of the interacting boson model 1. Furthermore, the structure of 182 Os was investigated under consideration of the influences of competing factors such as the nearby collective deformed region, γ-soft rotors, X(5) symmetry, and neighboring regions of shape coexistence in low excitation states. The study of higher-lying excitation bands in osmium isotopes and the associated isotones helped explain the structural transitions occurring at 182 Os, shedding light on the interplay of the different structural influences

Disorder and interference: localization phenomena

The specific problem we address in these lectures is the problem of transport and localization in disordered systems, when interference is present, as characteristic for waves, with a focus on realizations with ultracold atoms.Comment: Notes of a lecture delivered at the Les Houches School of Physics on "Ultracold gases and quantum information" 2009 in Singapore. v3: corrected mistakes, improved script for numerics, Chapter 9 in "Les Houches 2009 - Session XCI: Ultracold Gases and Quantum Information" edited by C. Miniatura et al. (Oxford University Press, 2011