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Dualities for Ising Networks.

Yu-Tin Huang1,2, Chia-Kai Kuo1, Congkao Wen3

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This study establishes a novel equivalence between planar Ising networks and the positive orthogonal Grassmannian. This connection enables efficient computation of Ising network correlators using recursive methods and renormalization group equations.

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Area of Science:

  • Statistical Mechanics
  • Algebraic Geometry
  • Computational Physics

Background:

  • Ising networks are fundamental models in statistical mechanics.
  • The positive orthogonal Grassmannian is a complex mathematical space.
  • Understanding their relationship is key to advancing computational methods.

Purpose of the Study:

  • To establish a formal equivalence between planar Ising networks and cells within the positive orthogonal Grassmannian.
  • To develop novel, efficient computational methods for Ising network correlators.
  • To explore the application of renormalization group equations in this context.

Main Methods:

  • Microscopic construction based on amalgamation to establish network-cell correspondence.
  • Recursive computation of correlators using duality moves.
  • Iterative amalgamation for logarithmic complexity scaling.

Main Results:

  • A direct correspondence is established for any planar Ising network.
  • Two recursive methods for computing Ising network correlators are introduced.
  • Fractal lattices emerge, with recursion formulas acting as exact renormalization group equations.

Conclusions:

  • The established equivalence provides a powerful framework for analyzing Ising networks.
  • The developed recursive methods offer significant computational advantages.
  • This work bridges statistical mechanics and algebraic geometry with practical computational benefits.