Height Fluctuations of Random Lozenge Tilings Through Nonintersecting Random Walks

Abstract

In this paper we study height fluctuations of random lozenge tilings of polygonal domains on the triangular lattice through nonintersecting Bernoulli random walks. For a large class of polygons which have exactly one horizontal upper boundary edge, we show that these random height functions converge to a Gaussian Free Field as predicted by Kenyon and Okounkov [28]. A key ingredient of our proof is a dynamical version of the discrete loop equations as introduced by Borodin, Guionnet and Gorin [5], which might be of independent interest.

arXiv comment70 pages, 10 figures. Draft version, comments are welcome!

References in corpus (6)

• A variational principle for domino tilings
• Asymptotics of uniformly random lozenge tilings of polygons. Gaussian free field
• Fluctuations of particle systems determined by Schur generating functions
• Fluctuations of eigenvalues of matrix models and their applications
• Lectures notes on compact Riemann surfaces
• Asymptotics of discrete $\beta$-corners processes via two-level discrete loop equations

Cited by in corpus (3)

• Lozenge tilings and the Gaussian free field on a cylinder
• Double Interlacing in Random Tiling Models
• Edge Statistics for Lozenge Tilings of Polygons, I: Concentration of Height Function on Strip Domains