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Finite-size corrections to disordered Ising models on random regular graphs.

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We found a formula for the first finite-size correction to the free energy of disordered Ising models on random regular graphs. This accounts for loop structures within the graph, impacting energy calculations.

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Area of Science:

  • Statistical Mechanics
  • Condensed Matter Physics
  • Graph Theory

Background:

  • Disordered Ising models are crucial for understanding magnetic materials.
  • Random regular graphs provide a tractable model for complex network structures.
  • Finite-size effects are essential for accurate theoretical predictions.

Purpose of the Study:

  • To derive an analytical expression for the first finite-size correction to the average free energy.
  • To provide a physical interpretation of this correction in terms of graph structures.
  • To advance the theoretical understanding of disordered systems on complex networks.

Main Methods:

  • Derivation of analytical expressions using techniques from statistical mechanics.
  • Analysis of disordered Ising models on random regular graphs.
  • Identification of loop structures and their contribution to free energy.

Main Results:

  • An exact formula for the first finite-size correction to the average free energy was obtained.
  • The formula is interpreted as a weighted sum over non-self-intersecting loops.
  • The weight is related to the free-energy shift from adding a loop to an infinite tree.

Conclusions:

  • The derived formula offers a new way to analyze finite-size effects in disordered systems.
  • The loop-based interpretation provides physical insight into the correction term.
  • This work contributes to the statistical mechanics of disordered systems on graphs.