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Finite-size scaling of a few statistical physics models in high dimensions Michta, Emmanuel


We investigate the behaviour of several classical models from statistical physics on a large but finite high-dimensional box. Our results elucidate in part the critical behaviour of the models as the volume of the box, its number of points, tends to infinity. The simplest model we study is the Simple Random Walk model for which we derive an exact asymptotic expansion of its massive two-point function. We obtain some new results but mostly unify the exposition and the proofs of the results we state, thereby hopefully making this presentation a point of reference for future works. For the weakly self-avoiding walk model, we prove that the walk on a torus behaves as on ℤᵈ provided that it has length much smaller than 𝑉¹⧸² where 𝑉 is the volume, number of points, of the torus which, we believe, is sharp. We also prove in that case that the corresponding scaling limit is Brownian motion on the torus. For high-dimensional percolation we first produce a useful estimate for the near-critical two-point function on ℤᵈ from which we can deduce that the torus two-point function has a plateau, i.e. that inside a critical window of parameters centered around the infinite-volume critical point, the torus two-point function decays as on ℤᵈ up until it reaches a constant value of order 𝑉⁻⁻²⧸³. Many other valuable results pertaining to finite-size scaling of the models are then deduced from the plateau. Finally, we present results for the hierarchical |φ|⁴ model in dimensions d ≥ 4 which identify exactly the critical behaviour of this model along with the role of boundary conditions in effective critical behaviour. This serves as a prototype for related models and thus leads to the formulation of precise conjectures for spin O(n) or SAW models in dimensions d ≥ 4. The results are derived from a rigorous Renormalisation Group analysis

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