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Published on: September 26, 2016
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Complete graph asymptotics for the Ising and random-cluster models on five-dimensional grids with a cyclic boundary
1Department of Mathematics and Mathematical Statistics, Umeå University, SE-901 87 Umeå, Sweden.
Summary
This study presents an exact scaling picture for the five-dimensional Ising model with cyclic boundary conditions. It resolves debates by comparing with the FK-random-cluster model and Monte Carlo simulations.
Area of Science:
- Statistical Mechanics
- Condensed Matter Physics
Background:
- Finite-size scaling behavior in five-dimensional Ising models remains debated.
- Previous studies relied on field theory or renormalization group methods.
Purpose of the Study:
- To propose a detailed and exact scaling picture for the critical region of the five-dimensional Ising model with cyclic boundary conditions.
- To resolve long-standing debates regarding its finite-size scaling behavior.
Main Methods:
- Comparison with exact and rigorous results for the FK-random-cluster model on a complete graph.
- Analysis of scaling regions in an L-dependent window around the critical point.
- Testing predictions against Monte Carlo simulation data.
Main Results:
- An exact scaling picture for the critical region of the Ising model with cyclic boundary is proposed.
- Distinct scaling regions are predicted and found to agree with simulation data.
- A bimodal energy distribution near the critical point, differing from the complete graph model, is identified.
Conclusions:
- The proposed scaling picture provides a rigorous framework for understanding the critical behavior of the five-dimensional Ising model.
- The identified bimodal energy distribution highlights a key difference between cyclic and free boundary conditions, potentially indicating a quasi-first-order phase transition.
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