Infinite density
Pro Research Analysisby 
Searched over 200M research papers for "infinite density"
Infinite Density: A Comprehensive Overview
Density Classification on Infinite Lattices and Trees
The concept of infinite density is explored in various contexts, including infinite graphs and cellular automata. One study addresses the density classification problem on infinite graphs where nodes are initially labeled by independent Bernoulli random variables. The goal is to design a system that evolves to determine whether the density parameter ( p ) is smaller or larger than 1/2. Solutions are provided for regular grids of dimension ( d > 1 ) and regular infinite trees, with numerical simulations proposed for the bi-infinite line .
Infinite Invariant Density in Semi-Markov Processes
In semi-Markov processes, infinite invariant density plays a crucial role. This density arises in models of anomalous diffusion, such as the generalized Lévy walk model. The state value tends to zero due to long interevent times, described by fat-tailed distributions. Two scaling laws describe the density accumulation near zero over time, providing universal behaviors and exact expressions for the infinite invariant density. These findings are significant for understanding the distribution of time-averaged observables in nonstationary processes .
Gravitational Instability and Infinite Density
In the context of gravitational instability, infinite density is achieved through the growth of perturbations during the expansion of matter without pressure. The solution involves unilateral compression leading to infinite density on disc-like surfaces, followed by adiabatic and shock wave compression of subsequent layers. This study provides an approximate yet qualitatively correct solution for large density perturbations .
Density Estimation in Infinite Dimensional Exponential Families
Infinite dimensional exponential families offer a rich framework for density estimation. These families can approximate a broad class of densities on ( \mathbb{R}^d ) using functions in a reproducing kernel Hilbert space. A proposed estimator minimizes the Fisher divergence between the unknown density and the family, providing consistent results and convergence rates under certain smoothness conditions. This method outperforms non-parametric kernel density estimators, especially as the dimensionality increases .
Subexponentiality of Densities in Infinitely Divisible Distributions
The subexponentiality of densities in infinitely divisible distributions is characterized by the equivalence of several properties, including the subexponentiality of the density and its Lévy measure. Under certain monotonic-type assumptions, these properties are shown to be equivalent, covering a wide class of distributions. This study also derives significant properties for analyzing subexponential densities, such as closure properties and factorization, which are useful for statistical inference .
Infinite Invariant Density and Weak Chaos
In weakly chaotic systems with marginal fixed points, infinite invariant density determines the statistics of time averages. Time averages of integrable observables follow the Aaronson-Darling-Kac theorem, while nonintegrable observables' time averages remain random. The distribution of these time averages is related to the infinite invariant density, with several identities established between amplitude ratios controlling the statistics .
Infinite Ergodic Theory for Heterogeneous Diffusion Processes
Infinite ergodic theory is applied to processes modeled by Langevin equations with multiplicative noise. A nonnormalized state, or infinite density, describes the statistical properties of these systems. The time averages of observables are obtained using ensemble averages with respect to the infinite density. The existence and shape of this density depend on the interpretation of the Langevin equation and the structure of the diffusion coefficient .
Infinite Density in Cold Atoms and Optical Lattices
Infinite densities also appear in the context of cold atoms in shallow optical lattices. These non-normalizable quasi-probability distributions describe long-time properties when ergodicity is broken. Semiclassical Monte Carlo simulations show that the momentum infinite density and its scale invariance should be observable in shallow potentials. The momentum autocorrelation function is evaluated in both stationary and nonstationary regimes .
Conclusion
The concept of infinite density spans various fields, from graph theory and semi-Markov processes to gravitational instability and quantum mechanics. Each study provides unique insights into how infinite density manifests and influences system behavior, offering valuable frameworks for further research and practical applications.
Sources and full results
Most relevant research papers on this topic