Description
- Abstract:
- In Chapter 1, we study planar ``vertex'' models, which are probability measures onedge subsets of a planar graph, satisfying certain constraints ateach vertex, examples including dimer model, and 1-2 model, which wewill define. We express the local statistics of a large class ofvertex models on a finite hexagonal lattice as a linear combinationof the local statistics of dimers on the corresponding Fisher graph,with the help of a generalized holographic algorithm. Using an$n\times n$ torus to approximate the periodic infinite graph, wegive an explicit integral formula for the free energy and localstatistics for configurations of the vertex model on an infinitebi-periodic graph. As an example, we simulate the 1-2 model by thetechnique of Glauber dynamics. In Chapter 2, we study the spectral curves of dimer models on periodic Fishergraphs, defined by the zero locus of the determinant of a modifiedweighted adjacency matrix. We prove that either they are disjointfrom the unit torus ($\mathbb{T}^2=\{(z,w):|z|=1,|w|=1\}$) or theyintersect $\mathbb{T}^2$ at a single real point. As an application, we prove that the single edge probability of dimer models on periodic Fisher graphs is unique under any translation invariant Gibbs measure. A periodic Ising model is one endowed with interactions that areinvariant under translations of members of a full-rank sublattice$\mathfrak{L}$ of $\mathbb{Z}^2$. In Chapter 3, we give an exact, quantitativedescription of the critical temperature, defined by the supreme ofthe temperatures at which the spontaneous magnetization of aperiodic, Ising ferromagnets is nonzero, as the solution of acertain algebraic equation, namely, the condition that the spectralcurve of the corresponding dimer model on the Fisher graph has areal node on the unit torus. A simple proof for the exponentialdecay of spin-spin correlations above the critical temperature forthe symmetric, periodic Ising ferromagnet, as well as theexponential decay of the edge-edge correlations for all non-criticaledge weights of the dimer model on periodic Fisher graphs, isobtained by our technique.
- Notes:
- Thesis (Ph.D. -- Brown University (2011)
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Citation
Li, Zhongyang,
"Vertex Models, Ising Models and Fisher Graphs"
(2011).
Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.7301/Z08P5XST
