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Liu, Yucheng

University of British Columbia

We investigate critical and near-critical scaling behaviours of various statistical mechanical models on the hypercubic lattice. Examples include the self-avoiding walk, the Ising model, Bernoulli bond percolation, and lattice trees and animals (branched polymers). In high-dimensional settings, we develop novel approaches to analyse the lace expansion equation, to derive mean-field critical behaviour for the two-point (correlation) functions. We also study the problem of finite-size scaling, and we develop a new general theory for the scaling of statistical mechanical models on a large box with periodic boundary conditions (torus). This thesis is divided into two parts. Part I focuses on the critical behaviour of two-point functions. For short-range (finite-range or exponentially decaying) models on the infinite lattice, we develop a new simple way to "deconvolve" the lace expansion equation, proving asymptotics for the critical two-point function. The method is also extended to models defined on the continuum. For models with long-range (polynomially decaying) couplings, we develop another method, based on an exact representation in terms of a random walk, that also yields critical behaviours. Our novel approaches significantly simplify previous analyses based on the lace expansion. Part II of the thesis is devoted to finite-size scaling under periodic boundary conditions. We develop a general theory that explains the "plateau" phenomenon on the torus: near the infinite-volume critical point, the torus two-point function does not decay to 0 but rather levels off to a constant called the torus plateau. The theory is based on a new, general comparison principle between the model on the torus and an effective "unwrapped" model on the infinite lattice, which we study using near-critical estimates obtained from significantly extending methods of Part I. The general theory stems from novel couplings for the Ising model and for lattice trees and animals, between the torus and the infinite lattice. Besides the two main parts, Chapter 2 of the thesis studies the weakly self-avoiding walk on the one-dimensional lattice. For a continuous-time version of the model, we prove a conjecture from 1993, that the escape speed of the weakly self-avoiding walk is strictly monotone in the parameter that controls self-repellence.

Text

2025-06-11

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/91332

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/