14 research outputs found
A cluster expansion approach to exponential random graph models
The exponential family of random graphs is among the most widely-studied
network models. We show that any exponential random graph model may
alternatively be viewed as a lattice gas model with a finite Banach space norm.
The system may then be treated by cluster expansion methods from statistical
mechanics. In particular, we derive a convergent power series expansion for the
limiting free energy in the case of small parameters. Since the free energy is
the generating function for the expectations of other random variables, this
characterizes the structure and behavior of the limiting network in this
parameter region.Comment: 15 pages, 1 figur
Dilatancy transition in a granular model
We introduce a model of granular matter and use a stress ensemble to analyze
shearing. Monte Carlo simulation shows the model to exhibit a second order
phase transition, associated with the onset of dilatancy.Comment: Future versions can be obtained from:
http://www.ma.utexas.edu/users/radin/papers/shear2.pd
Statistical mechanics of glass transition in lattice molecule models
Lattice molecule models are proposed in order to study statistical mechanics
of glass transition in finite dimensions. Molecules in the models are
represented by hard Wang tiles and their density is controlled by a chemical
potential. An infinite series of irregular ground states are constructed
theoretically. By defining a glass order parameter as a collection of the
overlap with each ground state, a thermodynamic transition to a glass phase is
found in a stratified Wang tiles model on a cubic lattice.Comment: 18 pages, 8 figure
Layering in crumpled sheets
We introduce a toy model of crumpled sheets, and use Monte Carlo simulation to show there is a first-order phase transition in the model, from a disordered dilute phase to a mixture with a layered phase. We demonstrate the transition through two order parameters, corr and lay, the first of which measures orientational order while the second measures bulk layering. An important feature of the argument is the behavior of the system as its size is increased