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Generation and Coherent Control of Pulsed Quantum Frequency Combs
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Published on: June 8, 2018

Crossing on hyperbolic lattices.

Hang Gu1, Robert M Ziff

  • 1Michigan Center for Theoretical Physics and Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109-2136, USA. ghbright@umich.edu

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|September 26, 2012
PubMed
Summary
This summary is machine-generated.

Researchers studied percolation crossing probabilities on hyperbolic lattices, finding that the probability increases with bond occupation. Critical percolation thresholds were estimated for heptagonal, pentagonal, and enhanced binary tree lattices.

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Area of Science:

  • Percolation theory
  • Statistical mechanics
  • Network science

Background:

  • Percolation theory studies connectivity in random networks.
  • Hyperbolic lattices offer unique geometric properties for network analysis.
  • Understanding critical phenomena in these lattices is crucial for various applications.

Purpose of the Study:

  • To investigate percolation crossing probabilities on specific hyperbolic lattices.
  • To determine the critical bond occupation probabilities (p_l and p_u) for these lattices.
  • To identify the self-duality point where crossing probability is 1/2.

Main Methods:

  • Dividing the circular boundary of hyperbolic lattices into four intervals.
  • Analyzing the probability of percolation crossing between opposite intervals.
  • Calculating bounds and estimates for critical percolation thresholds (p_l, p_u).
  • Identifying the self-duality point (p = 1/2 crossing probability).

Main Results:

  • Crossing probability increases monotonically from 0 to 1 as bond occupation probability (p) increases.
  • Estimated lower (p_l) and upper (p_u) critical values for heptagonal ({7,3}), enhanced or extended binary tree (EBT), EBT-dual, and pentagonal ({5,5}) lattices.
  • Identified the self-duality point for these lattices.

Conclusions:

  • The study provides critical insights into percolation phenomena on hyperbolic structures.
  • The findings contribute to the theoretical understanding of network connectivity and phase transitions.
  • Comparison with existing numerical and theoretical results validates the current estimations.