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Ising-like models on arbitrary graphs: the Hadamard way.

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We present a new framework for classical Ising-like models on graphs. This method uses the Hadamard transform to efficiently compute the energy spectrum, speeding up calculations for various graph types.

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Area of Science:

  • Statistical Mechanics
  • Graph Theory
  • Computational Physics

Background:

  • Classical Ising models are fundamental in statistical mechanics.
  • Understanding their behavior on complex graph structures is computationally challenging.
  • Existing methods for spectrum computation can be inefficient for large or arbitrary graphs.

Purpose of the Study:

  • To develop a generic and efficient framework for describing classical Ising-like models on arbitrary graphs.
  • To introduce a novel method for computing the energy spectrum of these models.
  • To explore potential computational speedups using established algorithms.

Main Methods:

  • Formulating a generic framework for Ising-like models on arbitrary graphs.
  • Representing the energy spectrum as a Hadamard transform of a sparse coding vector.
  • Applying the framework to regular graphs, specifically hypercubic graphs.
  • Developing a recurrence relation for spectrum determination.

Main Results:

  • The energy spectrum is shown to be the Hadamard transform of a sparse coding vector.
  • A simple recurrence relation for the spectrum was derived for regular graphs.
  • This approach significantly speeds up spectrum determination.
  • Initial analyses of partition functions and transfer matrices were conducted.

Conclusions:

  • The proposed framework offers an efficient method for computing the energy spectrum of classical Ising-like models.
  • The use of the Hadamard transform and sparse coding vectors has the potential to accelerate computations.
  • The derived recurrence relation provides a significant speedup for regular graph structures.
  • Further research into partition functions and transfer matrices is warranted.