arXiv subjects

Manuel Schrauth

Publications and source records attributed to Manuel Schrauth.

10 recordsLinked to original sources

arXiv2025

Hyperbolic tiling neighborhoods in O(1) time

Tilings of the hyperbolic plane are of significant interest among many branches of mathematics, physics and computer science. Yet, their construction remains a non-trivial task. Current approaches primarily use tree-based recursive algorithms, which are fundamentally limited: they do not readily yield the neighborhood graph representing cell adjacencies, which is however required for many applications. We introduce a novel approach that allows to build hyperbolic tilings and their associated graph structure simultaneously, using only combinatoric rules without requiring an explicit coordinate representation. This allows to generate arbitrarily large, exact hyperbolic graphs, with an algorithmic complexity that does not depend on the lattice size. We provide an easy-to-use implementation which substantially outperforms existing methods, hence rendering ultra large-scale numerical simulations on these geometric structures accessible for the scientific community.

physics.comp-ph↗

arXiv2024

HYPERTILING -- a high performance Python library for the generation and visualization of hyperbolic lattices

HYPERTILING is a high-performance Python library for the generation and visualization of regular hyperbolic lattices embedded in the Poincaré disk model. Using highly optimized, efficient algorithms, hyperbolic tilings with millions of vertices can be created in a matter of minutes on a single workstation computer. Facilities including computation of adjacent vertices, dynamic lattice manipulation, refinements, as well as powerful plotting and animation capabilities are provided to support advanced uses of hyperbolic graphs. In this manuscript, we present a comprehensive exploration of the package, encompassing its mathematical foundations, usage examples, applications, and a detailed description of its implementation.

physics.comp-ph↗

arXiv2022

Breitenlohner-Freedman bound on hyperbolic tilings

We establish how the Breitenlohner-Freedman (BF) bound is realized on tilings of two-dimensional Euclidean Anti-de Sitter space. For the continuum, the BF bound states that on Anti-de Sitter spaces, fluctuation modes remain stable for small negative mass-squared $m^2$. This follows from a real and positive total energy of the gravitational system. For finite cutoff $\varepsilon$, we solve the Klein-Gordon equation numerically on regular hyperbolic tilings. When $\varepsilon\to0$, we find that the continuum BF bound is approached in a manner independent of the tiling. We confirm these results via simulations of a hyperbolic electric circuit. Moreover, we propose a novel circuit including active elements that allows to further scan values of $m^2$ above the BF bound.

hep-th↗

arXiv2019

Universality of continuous phase transitions on random Voronoi graphs

The Voronoi construction is ubiquitous across the natural sciences and engineering. In statistical mechanics, though, critical phenomena have so far been only investigated on the Delaunay triangulation, the dual of a Voronoi graph. In this paper we set to fill this gap by studying the two most prominent systems of classical statistical mechanics, the equilibrium spin-1/2 Ising model and the non-equilibrium contact process, on two-dimensional random Voronoi graphs. Particular motivation comes from the fact that these graphs have vertices of constant coordination number, making it possible to isolate topological effects of quenched disorder from node-intrinsic coordination number disorder. Using large-scale numerical simulations and finite-size-scaling techniques, we are able to demonstrate that both systems belong to their respective clean universality classes. Therefore, quenched disorder introduced by the randomness of the lattice is irrelevant and does not influence the character of the phase transitions. We report the critical points of both models to considerable precision and, for the Ising model, also the first correction-to-scaling exponent.

cond-mat.stat-mech↗

arXiv2019

Relevance of topological disorder on the directed percolation phase transition

Despite decades of research, the precise role of topological disorder in critical phenomena has yet to be fully understood. A major contribution has been the work by Barghathi and Vojta, which uses spatial correlations to explain puzzling earlier results. However, due to its reliance on coordination number fluctuations, their criterion cannot be applied to constant-coordination lattices, raising the question, for which classes of transitions this type of disorder can be a relevant perturbation. In order to cast light on this question, we investigate the non-equilibrium phase transition of the two-dimensional contact process on different types of spatial random graphs with varying or constant coordination number. Using large-scale numerical simulations, we find the disorder to be relevant, as the dynamical scaling behaviour turns out to be non-conventional, thus ruling out the directed percolation universality class. We conjecture that the relevance of topological disorder is linked to how strongly connected the lattice is. Based on this assumption, we design two analysis tools that succeed in qualitatively distinguishing relevant from non-relevant topological disorders, supporting our conjecture and possibly pointing the way to a more complete relevance criterion.

cond-mat.stat-mech↗

arXiv2019

Fast algorithm for topologically disordered lattices with constant coordination number

We present a stochastic algorithm for constructing a topologically disordered (i.e., non-regular) spatial lattice with nodes of constant coordination number, the CC lattice. The construction procedure dramatically improves on an earlier proposal [Phys. Rev. E. 97, 022144 (2018)] with respect to both computational complexity and finite-size scaling properties - making the CC lattice an alternative to proximity graphs which, especially in higher dimensions, is significantly faster to build. Among other applications, physical systems such as certain amorphous materials with low concentration of coordination defects are an important example of disordered, constant-coordination lattices in nature. As a concrete application, we characterize the criticality of the 3D Ising model on the CC lattice. We find that its phase transition belongs to the clean Ising universality class, establishing that the disorder present in the CC lattice is a non-relevant perturbation in the sense of renormalization group theory.

cond-mat.dis-nn↗

arXiv2018

Diffusion and first-passage characteristics on a dynamically evolving support

We propose a generalized diffusion equation for a flat Euclidean space subjected to a continuous infinitesimal scale transform. For the special cases of an algebraic or exponential expansion/contraction, governed by time-dependent scale factors $a(t)\sim t^λ$ and $a(t)\sim \exp(μt)$, the partial differential equation is solved analytically and the asymptotic scaling behavior, as well as the dynamical exponents, are derived. Whereas in the algebraic case the two processes (diffusion and expansion) compete and a crossover is observed, we find that for exponential dynamics the expansion dominates on all time scales. For the case of contracting spaces, an algebraic evolution slows down the overall dynamics, reflected in terms of a new effective diffusion constant, whereas an exponential contraction neutralizes the diffusive behavior entirely and leads to a stationary state. Furthermore, we derive various first-passage properties and describe four qualitatively different regimes of (strong) recurrent/transient behavior depending on the scale factor exponent.

cond-mat.stat-mech↗

arXiv2018

A violation of the Harris-Barghathi-Vojta criterion

In 1974, Harris proposed his celebrated criterion: Continuous phase transitions in $d$-dimensional systems are stable against quenched spatial randomness whenever $dν>2$, where $ν$ is the clean critical exponent of the correlation length. Forty years later, motivated by violations of the Harris criterion for certain lattices such as Voronoi-Delaunay triangulations of random point clouds, Barghathi and Vojta put forth a modified criterion for topologically disordered systems: $aν>1$, where $a$ is the disorder decay exponent, which measures how fast coordination number fluctuations decay with increasing length scale. Here we present a topologically disordered lattice with coordination number fluctuations that decay as slowly as those of conventional uncorrelated randomness, but for which the universal behaviour is preserved, hence violating even the modified criterion.

cond-mat.dis-nn↗

arXiv2017

Two-dimensional Ising model on random lattices with constant coordination number

We study the two-dimensional Ising model on a network with a novel type of quenched topological (connectivity) disorder. We construct random lattices of constant coordination number and perform large scale Monte Carlo simulations in order to obtain critical exponents using finite-size scaling relations. We find disorder-dependent effective critical exponents, similar to diluted models, showing thus no clear universal behavior. Considering the very recent results for the two-dimensional Ising model on proximity graphs and the coordination number correlation analysis suggested by Barghathi and Vojta (2014), our results indicate that the planarity and connectedness of the lattice play an important role on deciding whether the phase transition is stable against quenched topological disorder.

cond-mat.stat-mech↗

arXiv2016

Characteristics of a random walk on a self-inflating support

Self-similar dynamical processes are characterized by a growing length scale $ξ$ which increases with time as $ξ\sim t^{1/z}$, where z is the dynamical exponent. The best known example is a simple random walk with z=2. Usually such processes are assumed to take place on a static background. In this paper we address the question what changes if the background itself evolves dynamically. As an example we consider a random walk on an isotropically and homogeneously inflating space. For an exponentially fast expansion it turns out that the self-similar properties of the random walk are destroyed. For an inflation with power-law characteristics, however, self-similarity is preserved provided that the exponent controlling the growth is small enough. The resulting probability distribution is analyzed in terms of cumulant ratios. Moreover, the dynamical exponent z is found to change continuously with the control exponent.

cond-mat.stat-mech↗