Work

Random surfaces and lattice Yang-Mills

Year: 2025

Type: article

Abstract: We study Wilson loop expectations in lattice Yang-Mills models with a compact Lie group G G . Using tools recently introduced in a companion paper (see M. Park, J. Pfeffer, S. Sheffield, and P. Yu [ Wilson loop expectations as sums over surfaces on the plane , 2023]), we provide alternate derivations, interpretations, and generalizations of several recent theorems about Brownian motion limits (Dahlqvist), lattice string trajectories (Chatterjee and Jafarov) and surface sums (Magee and Puder). We show further that one can express Wilson loop expectations as sums over embedded planar maps in a manner that applies to any matrix dimension N ≥ 1 N \geq 1 , any inverse temperature β > 0 \beta >0 , and any lattice dimension d ≥ 2 d \geq 2 . The surface expansions we use for this purpose have not (as far as we know) appeared in the literature before. When G = U ⁡ ( N ) G=\operatorname {U}(N) , the embedded maps we consider are pairs ( M , ϕ ) (\mathcal M, \phi ) where M \mathcal M is a planar (or higher genus) map and ϕ \phi is a graph homomorphism from M \mathcal M to a lattice such as Z d \mathbb Z^d . The faces of M \mathcal M come in two partite classes: edge-faces (each mapped by ϕ \phi onto a single edge) and plaquette-faces (ea

Source: Communications of the American Mathematical Society

DOI: 10.1090/cams/59

Authors: Sky Cao, Minjae Park, Scott Sheffield

Institutions: Massachusetts Institute of Technology, Auburn University

Language: English

Topic: Stochastic processes and statistical mechanics

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Funder: National Science Foundation

Awards: Random Surfaces and Related Questions, PostDoctoral Research Fellowship