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Related Concept Videos

Hyperbolas01:30

Hyperbolas

A hyperbola is a conic section produced when a double-napped cone is intersected by a plane at an angle steeper than the slope of the cone, such that it cuts through both nappes. This intersection yields two separate, mirror-image curves known as branches, which open away from each other along the transverse axis. The nearest points on each branch to the hyperbola’s center are termed vertices, and the distance from the center to a vertex is denoted by a. Perpendicular to the transverse axis is...
Hyperbolic Functions01:26

Hyperbolic Functions

A flexible cable suspended between two points at the same height naturally forms a curve known as a catenary. This shape results from the balance between the cable’s weight and the tension acting along its length, representing a state of mechanical equilibrium. Unlike simpler approximations, the true shape of a hanging cable is described using hyperbolic functions.Hyperbolic functions are closely related to exponential functions and are named for their connection to the geometry of the...
Hyperbolic and Inverse Hyperbolic Functions: Problem Solving01:30

Hyperbolic and Inverse Hyperbolic Functions: Problem Solving

An arched gate can be effectively modeled using a hyperbolic cosine profile because this type of function is smooth and symmetric about the vertical axis. When the arch is centered at the origin, its maximum height occurs at the center point. This symmetry ensures that any height below the crown of the arch is reached at two horizontal positions that are equal in distance from the centerline but lie on opposite sides.To determine where the gate reaches a height of five meters, the height of the...
Poisson's And Laplace's Equation01:25

Poisson's And Laplace's Equation

The electric potential of the system can be calculated by relating it to the electric charge densities that give rise to the electric potential. The differential form of Gauss's law expresses the electric field's divergence in terms of the electric charge density.
Bewley Lattice Diagram01:12

Bewley Lattice Diagram

The Bewley lattice diagram, developed by L. V. Bewley, effectively organizes the reflections occurring during transmission-line transients. It visually represents how voltage waves propagate and reflect within a transmission line, making it easier to understand the complex interactions that occur.
Geometry of Hyperbolas01:30

Geometry of Hyperbolas

A hyperbola consists of all points where the absolute difference of distances to two fixed points, called foci, remains constant. The standard equation isEach branch extends infinitely and approaches two asymptotes, which guide the curve’s behavior. The parameters a and b define key features: a measures the distance from the center to each vertex along the transverse axis, while b influences the slopes of the asymptotes. The asymptotes have equationsA rectangle centered at the origin with...

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Updated: Jun 25, 2026

Preparation of Free-Surface Hyperbolic Water Vortices
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Published on: July 28, 2023

Percolation on hyperbolic lattices.

Seung Ki Baek1, Petter Minnhagen, Beom Jun Kim

  • 1Department of Theoretical Physics, Umeå University, 901 87 Umeå, Sweden. garuda@tp.umu.se

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|March 5, 2009
PubMed
Summary
This summary is machine-generated.

Percolation transitions on hyperbolic lattices reveal two distinct thresholds. These findings suggest hyperbolic lattices form their own universality class, differing from Cayley trees in scaling properties.

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Area of Science:

  • Statistical Physics
  • Complex Systems
  • Network Science

Background:

  • Percolation theory studies the formation of connected clusters in random networks.
  • Hyperbolic lattices, with their diverging number of neighbors, present unique geometric properties.
  • Understanding phase transitions in these non-Euclidean structures is crucial for various scientific domains.

Purpose of the Study:

  • To numerically investigate percolation transitions on hyperbolic lattices.
  • To identify and characterize distinct percolation thresholds.
  • To compare the scaling properties of these transitions with those of the Cayley tree.

Main Methods:

  • Finite-size scaling analysis was employed for numerical simulations.
  • The study focused on identifying the emergence and behavior of unbounded clusters.
  • Critical exponents were calculated for the upper percolation threshold.

Main Results:

  • Two distinct percolation thresholds were confirmed on hyperbolic lattices.
  • A lower threshold exhibits scaling behavior similar to the Cayley tree.
  • An upper threshold shows unique scaling properties, distinct from the Cayley tree, with two critical exponents determined.

Conclusions:

  • Hyperbolic lattices exhibit unique percolation transition behavior.
  • These lattices likely form their own universality class for percolation.
  • The findings advance the understanding of phase transitions in complex, non-Euclidean networks.