arXiv subjects

Jens Grimm

Publications and source records attributed to Jens Grimm.

7 recordsLinked to original sources

arXiv2023

Two-point functions of random-length random walk on high-dimensional boxes

We study the two-point functions of a general class of random-length random walks on finite boxes in $\ZZ^d$ with $d\ge3$, and provide precise asymptotics for their behaviour. We show that the finite-box two-point function is asymptotic to the infinite-lattice two-point function when the typical walk length is $o(L^2)$, but develops a plateau when the typical walk length is $Ω(L^2)$. We also numerically study walk length moments and limiting distributions of the self-avoiding walk and Ising model on five-dimensional tori, and find that they agree asymptotically with the known results for self-avoiding walk on the complete graph, both at the critical point and also for a broad class of scaling windows/pseudocritical points. Furthermore, we show that the two-point function of the finite-box random-length random walk, with walk length chosen via the complete graph self-avoiding walk, agrees numerically with the two-point functions of the self-avoiding walk and Ising model on five-dimensional tori. We conjecture that these observations in five dimensions should also hold in all higher dimensions.

math-ph↗

arXiv2022

Unwrapped two-point functions on high-dimensional tori

We study unwrapped two-point functions for the Ising model, the self-avoiding walk and a random-length loop-erased random walk on high-dimensional lattices with periodic boundary conditions. While the standard two-point functions of these models have been observed to display an anomalous plateau behaviour, the unwrapped two-point functions are shown to display standard mean-field behaviour. Moreover, we argue that the asymptotic behaviour of these unwrapped two-point functions on the torus can be understood in terms of the standard two-point function of a random-length random walk model on Zd. A precise description is derived for the asymptotic behaviour of the latter. Finally, we consider a natural notion of the Ising walk length, and show numerically that the Ising and SAW walk lengths on high-dimensional tori show the same universal behaviour known for the SAW walk length on the complete graph.

math-ph↗

arXiv2020

Complete graph and Gaussian fixed point asymptotics in the five-dimensional Fortuin-Kasteleyn Ising model with periodic boundaries

We present an extensive Markov-chain Monte Carlo study of the finite-size scaling behavior of the Fortuin-Kasteleyn Ising model on five-dimensional hypercubic lattices with periodic boundary conditions. We observe that physical quantities, which include the contribution of the largest cluster, exhibit complete graph asymptotics. However, for quantities, where the contribution of the largest cluster is removed, we observe that the scaling behavior is mainly controlled by the Gaussian fixed point. Our results therefore suggest that \textit{both} scaling predictions, i.e. the complete graph \textit{and} the Gaussian fixed point asymptotics, are needed to provide a complete description for the five-dimensional finite-size scaling behavior on the torus.

cond-mat.stat-mech↗

arXiv2019

The length of self-avoiding walks on the complete graph

We study the variable-length ensemble of self-avoiding walks on the complete graph. We obtain the leading order asymptotics of the mean and variance of the walk length, as the number of vertices goes to infinity. Central limit theorems for the walk length are also established, in various regimes of fugacity. Particular attention is given to sequences of fugacities that converge to the critical point, and the effect of the rate of convergence of these fugacity sequences on the limiting walk length is studied in detail. Physically, this corresponds to studying the asymptotic walk length on a general class of pseudocritical points.

math-ph↗

arXiv2018

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

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.

cond-mat.stat-mech↗

arXiv2017

Lifted Worm Algorithm for the Ising Model

We design an irreversible worm algorithm for the zero-field ferromagnetic Ising model by using the lifting technique. We study the dynamic critical behavior of an energy estimator on both the complete graph and toroidal grids, and compare our findings with reversible algorithms such as the Prokof'ev-Svistunov worm algorithm. Our results show that the lifted worm algorithm improves the dynamic exponent of the energy estimator on the complete graph, and leads to a significant constant improvement on toroidal grids.

cond-mat.stat-mech↗

arXiv2017

Geometric explanation of anomalous finite-size scaling in high dimensions

We give an intuitive geometric explanation for the apparent breakdown of standard finite-size scaling in systems with periodic boundaries above the upper critical dimension. The Ising model and self-avoiding walk are simulated on five-dimensional hypercubic lattices with free and periodic boundary conditions, by using geometric representations and recently introduced Markov-chain Monte Carlo algorithms. We show that previously observed anomalous behaviour for correlation functions, measured on the standard Euclidean scale, can be removed by defining correlation functions on a scale which correctly accounts for windings.

cond-mat.stat-mech↗