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Random-Length Random Walks and Finite-Size Scaling in High Dimensions.

Zongzheng Zhou1, Jens Grimm1, Sheng Fang2

  • 1ARC Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS), School of Mathematical Sciences, Monash University, Clayton, Victoria 3800, Australia.

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This summary is machine-generated.

We rigorously prove that a random-length random walk model exhibits universal finite-size scaling behavior in high dimensions, matching conjectures for Ising and self-avoiding walk models.

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Area of Science:

  • Statistical Physics
  • High-Dimensional Systems
  • Mathematical Physics

Background:

  • The finite-size scaling (FSS) of the Ising model in high dimensions remains a debated topic.
  • Previous conjectures proposed universal FSS behavior for self-avoiding walks and Ising models in high dimensions.

Purpose of the Study:

  • To rigorously investigate the FSS behavior of the Ising model in high dimensions.
  • To introduce and analyze a novel random-length random walk model.

Main Methods:

  • Rigorous mathematical analysis of a random-length random walk model.
  • Extensive Monte Carlo simulations of the Ising model and self-avoiding walk.
  • Study on five-dimensional hypercubic lattices with free and periodic boundary conditions.

Main Results:

  • The random-length random walk model rigorously demonstrates universal FSS behavior.
  • This behavior aligns with previous conjectures for self-avoiding walks and Ising models.
  • The mean walk length of the random walk model dictates the scaling behavior of its Green's function.

Conclusions:

  • The study provides a rigorous foundation for understanding FSS in high dimensions.
  • Numerical simulations confirm the universality of the rigorous findings.
  • The random-length random walk model serves as a key to unlocking FSS universality in complex systems.