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Title:  Finite-size effects in high dimensional physical systems
Author:  Jens C. Grimm
As Contributor:   (not claimed)
Type: Ph.D.
Field: Physics
  • Mathematical Physics
Approach: Theoretical
Degree granting institution:  Monash University
Supervisor(s): Timothy M. Garoni
Defense date:  2018-06-29


This thesis consists of two parts. In the main part, we will study finite-size effects on high-dimensional physical systems. It is well-known that models of critical phenomena typically possess an upper critical dimension, $d_c$, such that in dimensions $d \ge d_c$, their thermodynamic behaviour is governed by critical exponents taking simple mean-field values. In contrast to the simplicity of the thermodynamic behaviour, the theory of finite-size scaling in dimensions above $d_c$ is surprisingly subtle, and remains the subject of ongoing debate. We address this long-standing debate, by introducing a \emph{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. In the second part, we will numerically compare the efficiency of various Markov-chain Monte Carlo algorithms for simulating the zero-field ferromagnetic Ising model. In particular, we will design an irreversible algorithm for the Ising model by using the lifting technique. Even though lifting is considered as a promising method to speed up Markov-chain Monte Carlo algorithms, it is an open question how it affects efficiency in specific examples. We will numerically study the dynamic critical behavior of an energy-like observable on both the complete graph and toroidal grids, and compare our findings with reversible worm algorithms. Our results show that the lifted algorithm improves the dynamic exponent of the energy-like observable on the complete graph, and leads to a significant constant improvement on high-dimensional toroidal grids.

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