Tootfinder

Dimension dependence of critical phenomena in long-range percolation
Tom Hutchcroft
https://arxiv.org/abs/2510.03951 https://arxiv.org/pdf/2510.03951

Dimension dependence of critical phenomena in long-range percolation
Statistical mechanical systems at and near their points of phase transition are expected to exhibit rich, fractal-like behaviour that is independent of the small-scale details of the system but depends strongly on the dimension in which the model is defined. Moreover, many models are conjectured to have an upper critical dimension with important quantitative and qualitative differences between critical behaviour at, above, and below the upper critical dimension. For models with long-range inter…

Hausdorff dimension of the Wedding cake type surfaces
Bal\'azs B\'ar\'any, Manuj Verma
https://arxiv.org/abs/2510.04613 https://arxiv.org/pdf/2…

Hausdorff dimension of the Wedding cake type surfaces
In this paper, we study the Hausdorff dimension of fractal interpolation surfaces (FISs) over a triangular domain. These FISs are known as `wedding cake surfaces'. These surfaces are the attractor of some deterministic self-affine iterated function systems (IFS) on $\mathbb{R}^3$ generated by a fractal interpolation algorithm. Due to the recent seminal result of Rapaport (Adv. Math. 449 (2024) 109734), the dimension theory of self-affine IFS on $\mathbb{R}^3$ is known whenever the IFS is strong…

Box dimension of the graphs of recurrent fractal interpolation functions
Lai Jiang, Xiao-Hui Li, Zhen Liang, Huo-Jun Ruan
https://arxiv.org/abs/2510.02754 https://

Box dimension of the graphs of recurrent fractal interpolation functions
Let $f$ be a generalized affine recurrent fractal interpolation function with vertical scaling functions. In this paper, by introducing underlying local iterated function systems of $f$, we define restricted vertical scaling matrices. Then we prove the monotonicity of spectral radii of these matrices without additional conditions. We also prove the irreducibility of these matrices under the assumption that vertical scaling functions are positive. With these results, we estimate the upper and lo…

Bilinear spherical maximal function with fractal dilations
Surjeet Singh Choudhary, Chun-Yen Shen, Saurabh Shrivastava
https://arxiv.org/abs/2510.03094 https://

Bilinear spherical maximal function with fractal dilations
In this paper, we investigate $L^p-$boundedness of the bilinear spherical maximal function associated with a general dilation set $E\subset\R_+$. We quantify the range of $L^p-$boundedness in terms of a dilation-invariant notion of upper Minkowski dimension of the set $E$. A particular case of this study settles an open question of $L^p-$boundedness of the lacunary bilinear spherical maximal function at borderline cases $p_1=1$ or $p_2=1$ in dimension $d\geq4$.

Control of single spin-flips in a Rydberg atomic fractal
Robin C. Verstraten, Ivo H. A. Knottnerus, Yu Chih Tseng, Alexander Urech, Tiago Santiago do Espirito Santo, Vinicius Zampronio, Florian Schreck, Robert J. C. Spreeuw, Cristiane Morais Smith
https://arxiv.org/abs/2509.03514

Control of single spin-flips in a Rydberg atomic fractal
Rydberg atoms trapped by optical tweezers have emerged as a versatile platform to emulate lattices with different geometries, in which long-range interacting spins lead to fascinating phenomena, ranging from spin liquids to topological states of matter. Here, we show that when the lattice has a fractal geometry with Hausdorff dimension 1.58, additional surprises appear. The system is described by a transverse-field Ising model with long-range van der Waals interactions in a Sierpinski gasket fr…

An Approach to Estimating Quadratic Logistic Model Parameters of Fractal Dimension Curves
Yanguang Chen
https://arxiv.org/abs/2508.19917 https://arxiv.org/…

An Approach to Estimating Quadratic Logistic Model Parameters of Fractal Dimension Curves
The fractal dimension curves of urban form and growth fall into two categories: One can be described by common logistic function, and the other can be described with quadratic logistic function. The approach to estimating the parameter of the ordinary logistic model has been developed. However, how to estimate the parameter of quadratic logistic model is still a problem. This paper is devoted to finding a nonlinear regressive approach for estimating parameter values of quadratic logistic model …

Strichartz and local smoothing estimates for the fractional Schr\"odinger equations over fractal time
Jin Bong Lee, Sanghyuk Lee, Luz Roncal
https://arxiv.org/abs/2508.16102

Strichartz and local smoothing estimates for the fractional Schrödinger equations over fractal time
We obtain Strichartz-type estimates for the fractional Schrödinger operator $f \mapsto e^{it(-Δ)^{γ/2}} f$ over a time set $E$ of fractal dimension. To obtain those estimates capturing fractal nature of $E$, we employ the notions in the spirit of the Assouad dimension, such as, bounded Assouad characteristic and Assouad specturm. We also prove the estimate $$ \| e^{it(-Δ)^{γ/2}} f \|_{L_t^q(\mathrm{d}μ; L_x^r(\mathbb{R}^d))} \le C \|f\|_{H^s}, $$ where $μ$ is a measur…

Fractal analysis of slow-fast and regular systems: A survey of recent results and future perspectives
Renato Huzak (Hasselt University), Goran Radunovi\'c (University of Zagreb, Faculty of Science), Vesna \v{Z}upanovi\'c (University of Zagreb, Faculty of Electrical Engineering and Computing)
https://arxiv.org/abs/2508.19859

Fractal analysis of slow-fast and regular systems: A survey of recent results and future perspectives
We survey recent developments in fractal analysis of regular and slow-fast dynamical systems using Minkowski dimension. Our focus is on spiral trajectories near monodromic limit periodic sets in regular systems and entry-exit sequences in slow-fast systems with degenerate singularities. For regular systems, we recall connections between Minkowski dimension and cyclicity. Key results include fractal classifications of weak foci, degenerate foci, and polycycles, where dimensional relationships pr…

Fractal depth-first search paths in statistical physics models
Qiyuan Shi, Youjin Deng, Ming Li
https://arxiv.org/abs/2508.10502 https://arxiv.org/pdf/2508…

Fractal depth-first search paths in statistical physics models
We study the fractal properties of depth-first search (DFS) paths in critical configurations of statistical physics models, including the two-dimensional $O(n)$ loop model for various $n$, and bond percolation in dimensions $d = 2$ to $6$. In the $O(n)$ loop model, the fractal dimension of the DFS path consistently matches that of the external perimeter, suggesting DFS as a convenient alternative for probing interface geometry. For bond percolation, the DFS path exhibits nontrivial fractal scal…

Holographic Turbulence and the Fractal Dimension of the Turbulent Horizon
Jia Du, Yu Tian, Hongbao Zhang
https://arxiv.org/abs/2510.12198 https://arxiv.org…

Holographic Turbulence and the Fractal Dimension of the Turbulent Horizon
We study two-dimensional turbulence driven by a scalar operator within the framework of the AdS/CFT correspondence, where the external driving source is used to sustain a quasi-steady turbulent state. We numerically construct dynamical and spatially inhomogeneous turbulent black holes in the asymptotically $\mathrm{AdS}_4$ spacetime by solving the full nonlinear equations of motion in the Bondi-Sachs formalism. The inverse energy cascade and the corresponding energy spectrum of both decaying an…

Quasisymmetric mappings on two variants of fractal percolation
Roope Anttila, Sylvester Eriksson-Bique, Aleksi Py\"or\"al\"a
https://arxiv.org/abs/2510.06900 http…

Quasisymmetric mappings on two variants of fractal percolation
We study quasisymmetric maps on two variants of the classical fractal percolation model: the fat and dense fractal percolations. We show that, almost surely conditioned on non-extinction, the Hausdorff dimension of the fat fractal percolation cannot be lowered with a quasisymmetry and the Hausdorff dimension of the dense fractal percolation cannot be lowered with a power quasisymmetry.

Delocalization Induced by Enhanced Hyperuniformity in One-Dimensional Disordered Systems
Junmo Jeon, Harukuni Ikeda, Shiro Sakai
https://arxiv.org/abs/2509.23036 https://…

Delocalization Induced by Enhanced Hyperuniformity in One-Dimensional Disordered Systems
In one dimension, any disorder is traditionally believed to localize all states. We show that this paradigm breaks down under hyperuniform disorder, which suppresses long-wavelength fluctuations and interpolates between random and periodic potentials. In tight-binding chains, strong hyperuniformity induces a sharp delocalization transition and the emergence of mobility edges. The transition is identified by the generalized fractal dimension and corroborated by the scaling of localization length…

The dimension of well approximable numbers
Victor Beresnevich, Sanju Velani
https://arxiv.org/abs/2509.05687 https://arxiv.org/pdf/2509.05687

The dimension of well approximable numbers
In this survey article, we explore a central theme in Diophantine approximation inspired by a celebrated result of Besicovitch on the Hausdorff dimension of well approximable real numbers. We outline some of the key developments stemming from Besicovitch's result, with a focus on the Mass Transference Principle, Ubiquity and Diophantine approximation on manifolds and fractals. We highlight the subtle yet profound connections between number theory and fractal geometry, and discuss several open p…

Fractality of open clusters in singles, pairs, and groups
Almat Akhmetali
https://arxiv.org/abs/2508.13980 https://arxiv.org/pdf/2508.13980

Fractality of open clusters in singles, pairs, and groups
In this work, we investigate the global structural properties and fractality of 1,876 open clusters (OCs) in different environments, including 1,145 singles, 392 pairs, and 339 groups. We analyze cluster mass, age, size, concentration, and fractal structure using the Q parameter and the fractal dimension fdim, and examine their correlations with key physical parameters. Our results reveal systematic environmental trends: clusters in groups are generally younger, less massive, slightly larger, a…

Recent Progress on Fractal Percolation
Istv\'an Kolossv\'ary, Sascha Troscheit
https://arxiv.org/abs/2508.08150 https://arxiv.org/pdf/2508.08150

Recent Progress on Fractal Percolation
This is a survey paper about the fractal percolation process, also known as Mandelbrot percolation. It is intended to give a general breadth overview of more recent research in the topic, but also includes some of the more classical results, for example related to the connectivity properties. Particular emphasis is put on the dimension theory of the limiting set and also on the geometry of the non-trivial connected components in the supercritical regime. In particular, we show that both the Ass…

Replaced article(s) found for cond-mat.str-el. https://arxiv.org/list/cond-mat.str-el/new
[1/1]:
- Fractal dimension and the counting rule of the Goldstone modes
Qian-Qian Shi, Yan-Wei Dai, Huan-Qiang Zhou, Ian P. McCulloch

Generalized Thue-Morse measures: spectral and fractal analysis
Philipp Gohlke, Marc Kesseb\"ohmer, Tanja I. Schindler
https://arxiv.org/abs/2509.22109 https://

Generalized Thue-Morse measures: spectral and fractal analysis
We investigate a family of Riesz products and show that they can be regarded as diffraction measures of generalized Thue-Morse sequences, possibly over an infinite alphabet. These measures are closely related to the dynamical system arising from the doubling map together with an observable exhibiting a logarithmic singularity. For this system, we develop a generalized thermodynamic formalism beyond the standard setting, which yields explicit formulas for Birkhoff and dimension spectra. A furthe…

Localization Transition on Random Graphs with Chiral and Bogoliubov-de Gennes Symmetry Classes
Daniil Kochergin
https://arxiv.org/abs/2510.10255 https://ar…

Localization Transition on Random Graphs with Chiral and Bogoliubov-de Gennes Symmetry Classes
We studied single-particle Anderson localization in ensembles of graphs that correspond to chiral and Bogoliubov-de Gennes (BdG) symmetry classes. For a random biregular bipartite graph with chiral symmetry, the density of states was found using the cavity approach. Calculating the fractal dimension shows the effects of disordered zero modes. For Bogoliubov-de Gennes ensembles with an underlying random regular graph (RRG), the density of states was calculated both numerically and analytically. …