tailieunhanh - Báo cáo toán học: "On the Dimer Problem and the Ising Problem in Finite 3-dimensional Lattices"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: On the Dimer Problem and the Ising Problem in Finite 3-dimensional Lattices. | On the Dimer Problem and the Ising Problem in Finite 3-dimensional Lattices Martin Loebl Department of Applied Mathematics and Institute for Theoretical Computer Science ITI Charles University Malostranske n. 25 118 00 Praha 1 Czech Republic loebl@ Submitted April 11 2001 Accepted July 8 2002 MR Subject Classifications 05B35 05C15 05A15 Abstract We present a new expression for the partition function of the dimer arrangements and the Ising partition function of the 3-dimensional cubic lattice. We use the Pfaffian method. The partition functions are expressed by means of expectations of determinants and Pfaffians of matrices associated with the cubic lattice. 1 Introduction The close-packed dimer model of statistical mechanics can be stated as follows. One considers a set of sites and a set of bonds connecting certain pairs of sites. Each bond b can absorb a dimer which represents a diatomic molecule with corresponding energy Eb. It is required that each site is occupied exactly once by one of the atoms of a dimer. A state s is an arrangement of dimers which meets this requirement and its energy E s is 52 Eb where the sum is taken over all bonds b which absorb a dimer. Then the partition function of the dimer model may be viewed as a density function of energy levels. The dimer model was first considered by Roberts 18 in 1935 and by Fowler and Rushbrook 3 . The dimer model for 2-dimensional lattices appears in calculations of the thermodynamic properties of a system of diatomic molecules-dimers. It has been solved Partially supported by the Project LN00A056 of the Czech Ministry of Education by GAUK 158 grant and by FONDAP on applied mathematics THE ELECTRONIC JOURNAL OF COMBINATORICS 9 2002 R30 1 by Kasteleyn 13 and by Temperley and Fisher 11 . The same problem for 3-dimensional lattices remains an important open problem of statistical physics see 14 for references . Many fundamental observations about the dimer and monomer-dimer model .