# Random-length Random Walks and Finite-size Scaling on high-dimensional hypercubic lattices I: Periodic Boundary Conditions

@article{Zhou2020RandomlengthRW, title={Random-length Random Walks and Finite-size Scaling on high-dimensional hypercubic lattices I: Periodic Boundary Conditions}, author={Zongzheng Zhou and Jens Grimm and Youjin Deng and Timothy M. Garoni}, journal={arXiv: Mathematical Physics}, year={2020} }

We study a general model of random-length random walks on discrete tori, and show that the mean walk length controls the scaling of the two-point function. We conjecture that on tori of dimension at least 5, the two-point functions of the Ising model and self-avoiding walk display the same scaling as the random-length random walk, not only at criticality, but also for a broad class of scaling windows/pseudocritical points. This conjecture is supported by extensive Monte Carlo simulations of the…

## 9 Citations

Unwrapped two-point functions on high-dimensional tori

- Mathematics, Physics
- 2022

We study unwrapped two-point functions for the Ising model, the selfavoiding walk and a random-length loop-erased random walk on high-dimensional lattices with periodic boundary conditions. While the…

The length of self-avoiding walks on the complete graph

- MathematicsJournal of Statistical Mechanics: Theory and Experiment
- 2019

The leading order asymptotics of the mean and variance of the walk length are obtained, as the number of vertices goes to infinity, in the variable-length ensemble of self-avoiding walks on the complete graph.

The near-critical two-point function and the torus plateau for weakly self-avoiding walk in high dimensions

- Mathematics
- 2020

We use the lace expansion to study the long-distance decay of the two-point function of weakly self-avoiding walk on the integer lattice Z in dimensions d > 4, in the vicinity of the critical point,…

The near-critical two-point function for weakly self-avoiding walk in high dimensions

- Mathematics
- 2020

We use the lace expansion to study the long-distance decay of the two-point function of weakly self-avoiding walk on the integer lattice $\mathbb{Z}^d$ in dimensions $d>4$, in the vicinity of the…

The scaling limit of the weakly self-avoiding walk on a high-dimensional torus

- Mathematics
- 2021

We prove that the scaling limit of the weakly self-avoiding walk on a d-dimensional discrete torus is Brownian motion on the continuum torus if the length of the rescaled walk is o(V ) where V is the…

Nonreversible Markov Chain Monte Carlo Algorithm for Efficient Generation of Self-Avoiding Walks

- MathematicsFrontiers in Physics
- 2021

An efficient nonreversible Markov chain Monte Carlo algorithm to generate self-avoiding walks with a variable endpoint that allows for three types of elementary moves on the existing self- avoidanceing walk: shorten, extend or alter conformation without changing the length of the walk.

High-dimensional near-critical percolation and the torus plateau

- Mathematics
- 2021

We consider percolation on Z and on the d-dimensional discrete torus, in dimensions d ≥ 11 for the nearest-neighbour model and in dimensions d > 6 for spread-out models. For Z, we employ a wide range…

Anomalous finite-size scaling in the Fortuin-Kasteleyn clusters of the five-dimensional Ising model with periodic boundary conditions

- Physics
- 2019

We present a Monte Carlo study of the Fortuin-Kasteleyn Ising model on a five-dimensional ($d=5$) hypercubic lattice with linear size $L$ and periodic boundary conditions. Our numerical results show…

Ising Model with Curie–Weiss Perturbation

- MathematicsJournal of Statistical Physics
- 2022

. Consider the nearest-neighbor Ising model on Λ n := [ − n, n ] d ∩ Z d at inverse temperature β ≥ 0 with free boundary conditions, and let Y n ( σ ) := P u ∈ Λ n σ u be its total magnetization. Let…

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The near-critical two-point function for weakly self-avoiding walk in high dimensions

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We use the lace expansion to study the long-distance decay of the two-point function of weakly self-avoiding walk on the integer lattice $\mathbb{Z}^d$ in dimensions $d>4$, in the vicinity of the…

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