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Correlation inequalities of GKS type for the Potts model
Correlation inequalities are presented for ferromagnetic Potts models with external field, using the random cluster representation of Fortuin and Kasteleyn, together with the FKG inequality. These results extend and simplify ...
The work of Lucio Russo on percolation
The contributions of Lucio Russo to the mathematics of percolation and disordered systems are outlined. The context of his work is explained, and its ongoing impact on current work is described and amplified.
The 1-2 model
(American Mathematical SocietyIn the Tradition of Ahlfors–Bers, VII, 2017)
The current paper is a short review of rigorous results for the 1-2 model. The 1-2 model on the hexagonal lattice is a model of statistical mechanics in which each vertex is constrained to have degree either 1 or 2. It was ...
Critical Surface of the Hexagonal Polygon Model
(SpringerJournal of Statistical Physics, 2016-05-01)
The hexagonal polygon model arises in a natural way via a transformation of the 1-2 model on the hexagonal lattice, and it is related to the high temperature expansion of the Ising model. There are three types of edge, and ...
Connective constants and height functions for Cayley graphs
(American Mathematical SocietyTransactions of the American Mathematical Society, 2017-03-31)
The connective constant $μ$($G$) of an infinite transitive graph $G$ is the exponential growth rate of the number of self-avoiding walks from a given origin. In earlier work of Grimmett and Li, a locality theorem was proved ...
Critical surface of the 1-2 model
The 1-2 model on the hexagonal lattice is a model of statistical mechanics in which each vertex is constrained to have degree either 1 or 2. There are three edge directions, and three corresponding parameters a, b, c. It ...