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.

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.Finite-size Scaling (FSS) is a fundamental physical theory within statistical mechanics, describing the asymptotic approach to the thermodynamic limit of finite systems in the neighbourhood of a critical phase transi-It is well-known [3] that models of critical phenomena typically possess an upper critical dimension, d c , such that in dimensions d ≥ d c , their thermodynamic behaviour is governed by critical exponents taking simple mean-field values [4]. In contrast to the simplicity of the thermodynamic behaviour, however, the theory of FSS in dimensions above d c is surprisingly subtle, and remains the subject of ongoing debate [5][6][7][8][9][10][11][12]. We will show here that such subtleties can be explained in a simple way, by taking an appropriate geometric perspective.Perhaps the most important class of models in equilibrium statistical mechanics are the n-vector models [13], describing systems of pairwise-interacting unit-vector spins in R n [14]. The cases n = 1, 2, 3 respectively correspond to the Ising, XY and Heisenberg models of ferromagnetism, while the limiting case n = 0 corresponds to the Self-avoiding Walk (SAW) model of polymers [3].The n-vector model has wide-ranging applications in condensed matter physics, particularly in the theory of superfluidity/superconductivity and quantum magnetism. In addition, the case n = 2 is related to the Bose-Hubbard model [15] which is actively studied in the field of ultra-cold atom physics. In such quantum applications, the quantum system in d spatial dimensions is related to the classical model in d + 1 dimensions. Since [3] d c = 4 for the nearest-neighbour n-vector model, this shows that understanding its FSS when d ≥ d c is of importance not only to the theory of FSS itself, but also in the field of condensed matter physics more generally. We also note that the value of d c can be reduced by the introduction of long-range interactions.

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.

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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 energylike observable 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 energylike observable on the complete graph and leads to a significant constant improvement on toroidal grids.

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We study unwrapped two-point functions for the Ising model, the self-avoiding walk (SAW) 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 Z d . 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.