We study cluster algebras with principal and arbitrary coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of certain paths on a triangulation of the surface. As an immediate consequence, we prove the positivity conjecture of Fomin and Zelevinsky for these cluster a...

The renormalization group (RG) approach is largely responsible for the considerable success which has been achieved in developing a quantitative theory of phase transitions. This work treats the rigorous definition of the RG map for classical Ising-type lattice systems. A cluster expansion is used to justify this definition in the infinite volume limit at high temperature.

We study statistical mechanics of the self–gravitating system applying the cluster expansion method developed in solid state physics. By summing infinite series of diagrams, we derive a complex free energy whose imaginary part is related to the relaxation time of the system. Summation of another series yields two–point correlation function whose correlation length is essentially given by the Je...

We summarize results of [1]. Among other things, they compute the first order expansion to the surface tension for all q−state Potts models at very low temperatures. Using this expansion, they determine the asymptotic shape of the Wulff crystal defined by the surface tension functional determined by an infinite volume Gibbs State at very low temperature. Their results are formulated in terms of...

M. K. Y. Chan,1,2 J. Reed,3 D. Donadio,4 T. Mueller,2 Y. S. Meng,2,5 G. Galli,4 and G. Ceder2 1Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 2Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 3Lawrence Livermore National Laboratory, Livermore, CA 94551, USA 4Department of Chemistry, Universit...

as an empirical formula expressing P in a power series of particle density ρ = N/V , where N is the number of particles. A first-principle understanding of the virial expansion was provided years later by the advent of the Mayer cluster expansion in statistical mechanics in the 1930s. However, following Onnes the virial expansion has since been generally regarded as an expansion in density. Her...