Limit theorems for Ginzburg-Landau [grad phi] random surfaces
- We study the massless field on lattice approximations of planar domains with smooth boundary, with uniformly convex, nearest neighbor, gradient interaction. This is the so-called Ginzburg-Landau (GL) interface model. It is a general model for a (2+1)-dimensional effective interface in which h represents the height. Our main results are two new limit theorems for functionals of h: (CLT) linear functionals of the height converge to the continuum Gaussian free field and the macroscopic level sets converge to variants of the Schramm-Loewner evolution (SLE). The main step in the proof of both results is a new estimate which describes how the boundary data of the field affects its law in the bulk. Along the way, we also give a new construction of the shift ergodic GL Gibbs (Funaki-Spohn) states. For the CLT, we take our boundary conditions to be a continuous perturbation of a macroscopic tilt. We prove that the fluctuations of linear functionals of h about the tilt converge in the limit to a Gaussian free field on D, the standard Gaussian with respect to a weighted Dirichlet inner product whose weights we will identify explicitly. The limiting result for the level sets resolves a conjecture of Sheffield that SLE(4), a conformally invariant random curve, is the universal limit of the chordal zero-height contours of random surfaces with isotropic, uniformly convex potentials.
|Type of resource
|electronic; electronic resource; remote
|1 online resource.
|Miller, Jason Peter
|Stanford University, Department of Mathematics
|Statement of responsibility
|Jason Peter Miller.
|Submitted to the Department of Mathematics.
|Thesis (Ph.D.)--Stanford University, 2011.
- © 2011 by Jason Peter Miller
- This work is licensed under a Creative Commons Attribution 3.0 Unported license (CC BY).
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