Ferromagneticity of simplicial fields on two-dimensional compact manifolds
Sergio A Albeverio, ERLING G.B. HOHLER, Bogusław Zegarliński · Journal of Mathematical Physics · 1992
Smooth triangulations of a compact smooth connected two-dimensional Riemannian manifold M are considered. The q-simplicial fields are defined with values in the space of q-cochains and a natural Gaussian measure is defined giving their distribution, with covariance defined essentially in terms of the combinatorial Laplacian Δc. In the continuum limit this measure for q=0 is the free quantum field measure over M. In this case it is shown that for a certain collection of triangulations there exists a sequence of subdivisions of each triangulation such that the corresponding measure is ferromagnetic. It is also shown that for sufficiently fine subdivisions −Δ+m2I, m≳0 has nonpositive off-diagonal elements. The proofs are obtained by a result on triangulations by simplexes with acute angles. It is also proven that the probability measures describing quantum fields on M with polynomial, trigonometric, or exponential interactions satisfy FKG inequalities.