Charles Radin

Space tilings and local isomorphism

We prove for a large class of tilings that, given a finite tile set, if it is possible to tile Euclideann-space with isometric copies of this set, then there is a tiling with the ‘local isomorphism property’.

The ground state for soft disks

We consider some two-dimensional models of point particles interacting through short-range two-body potentials and prove that their zero temperature, zero pressure states are crystalline.

LOW TEMPERATURE AND THE ORIGIN OF CRYSTALLINE SYMMETRY

This is a status report on the classical problem of determining the origins of crystalline symmetry in low temperature matter.

The Ground State for Sticky Disks

It is proven that the ground state of the two-dimensional sticky potential is the triangular lattice.

Phase transitions in exponential random graphs.

We derive the full phase diagram for a large family of two-parameter exponential random graph models, each containing a first order transition curve ending in a critical point.

Space tilings and substitutions

We generalize the study of symbolic dynamical systems of finite type and ℤ2 action, and the associated use of symbolic substitution dynamical systems, to dynamical systems with ℝ2 action. The new systems are associated with tilings of the plane. We generalize the classical technique of the matrix of a substitution to include the geometrical information needed to study tilings, and we utilize rotation invariance to eliminate discrete spectrum. As an example we prove that the pinwheel tilings have no discrete spectrum.

The infinite-volume ground state of the Lennard-Jones potential

We consider a finite chain of particles in one dimension, interacting through the Lennard-Jones potential. We prove the ground state is unique, and approaches uniform spacing in the infinite-particle limit.

Random Close Packing of Granular Matter

We propose an interpretation of the random close packing of granular materials as a phase transition, and discuss the possibility of experimental verification.

Tiling, periodicity, and crystals

A result on nonperiodic tilings is generalized and related to the problem of the origin of crystalline symmetry.

Phase transitions in a complex network

We study a mean field model of a complex network, focusing on edge and triangle densities. Our first result is the derivation of a variational characterization of the entropy density, compatible with the infinite node limit. We then determine the optimizing graphs for small triangle density and a range of edge density, though we can only prove they are local, not global, maxima of the entropy density. With this assumption we then prove that the resulting entropy density must lose its analyticity in various regimes. In particular this implies the existence of a phase transition between distinct heterogeneous multipartite phases at low triangle density, and a phase transition between these phases and the disordered phase at high triangle density.