A central limit theorem for lattice gauge gibbs random fields

Abstract

Seiler [5] introduced the lattice gauge theories, in which the discrete gauge fields theories were described by the lattice Gibbs random fields model. For the lattice gauge theories with finite abelian group G in the weak coupling regime, it was shown that there is only one translation invariant Gibbs state in the infinite volume[l]. This paper discussed the central limit theorem for energy variables under this Gibbs state. The previous sufficient conditions of the central limit theorem for energy variables [4] were checked with dlfBculty in the lattice gauge theories. The notations are similar to those of [1].. Denote by ~ the three-dimensional integer lattice, and by G the finite abelian group. Two nearest points in Z a are called a llnl¢~ and a square in Z 3 with unit length is called a plaquette. Pick a positive orientation for the tinl~s and plaquette. If ~', t* G Z 3 are two nearest points and the positive orientation is from g to ~, denote by (gt~ the ]inl~. Let ~ be the set of all positive links in Z 3. Let A be a three-dimensional box in Z a, and denote by Al and Ap the sets of llnlcs and plaquettes in A respectively. A map ~/ : ~ ~ ~ G is called a gauge field configuration. If (~*~ G ~ , define 71((~*~) = ~7 -1 ((gt-)), where ~1-1(. ) is the inverse element of ~7(" ) in G. If the positive orientation of plaquette p is 8"1 "+ 82 "-+ 8*a -'+ o'4, define ~(p) : = ~((8283)) " ~7((~18"2)) "~/((g394)) • ~((g4gl)), where "." is the product in G. Let the map f : G , ~ ~ satisfy the following conditions: 1) f (aba -1) = f(b), a, b G G; 2) the unit element e E G is the unique maximum point of f . Now we can define the Hamiltonian of the lattice gauge field on box A C ~ by