Random-length Random Walks and Finite-size scaling in high dimensions

Abstract

We address a long-standing debate regarding the finite-size scaling of the Ising model in high dimensions, by introducing a random-length random walk model, which we then study rigorously. We prove that this model exhibits the same universal FSS behaviour previously conjectured for the self-avoiding walk and Ising model on finite boxes in high-dimensional lattices. Our results show that the mean walk length of the random walk model controls the scaling behaviour of the corresponding Green's function. We numerically demonstrate the universality of our rigorous findings by extensive Monte Carlo simulations of the Ising model and self-avoiding walk on five-dimensional hypercubic lattices with free and periodic boundaries.

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Cited by in corpus (20)

• Extraordinary-log surface phase transition in the three-dimensional $XY$ model
• Finite-size Scaling of O($n$) Systems at the Upper Critical Dimensionality
• Complete graph and Gaussian fixed point asymptotics in the five-dimensional Fortuin-Kasteleyn Ising model with periodic boundaries
• Geometric Upper Critical Dimensions of the Ising Model
• Geometric scaling behaviors of the Fortuin-Kasteleyn Ising model in high dimensions
• The length of self-avoiding walks on the complete graph
• Quantum extraordinary-log universality of boundary critical behavior
• Weakly self-avoiding walk on a high-dimensional torus
• Classical-quantum correspondence of special and extraordinary-log criticality: Villain's bridge
• Logarithmic finite-size scaling of the self-avoiding walk at four dimensions
• Percolation effects in the Fortuin-Kasteleyn Ising model on the complete graph
• Unwrapped two-point functions on high-dimensional tori
• Effective-Dimension Theory of Critical Phenomena above Upper Critical Dimensions
• Ising model with Curie-Weiss perturbation
• Finite-Size Scaling of the High-Dimensional Ising Model in the Loop Representation
• Two-point functions of random-length random walk on high-dimensional boxes
• A general approach to massive upper bound for two-point function with application to self-avoiding walk torus plateau
• Logarithmic Finite-Size Scaling of the Four-Dimensional Ising Model
• Exact solution of weighted partially directed walks crossing a square
• The torus plateau for the high-dimensional Ising model