Transfer matrix computation of critical polynomials for twodimensional Potts models
We showed, In our previous work, that critical manifolds of the qstate Potts model can be studied by means of a graph polynomial P _{B}(q, v), henceforth referred to as the critical polynomial. This polynomial may be defined on any periodic twodimensional lattice. It depends on a finite subgraph B, called the basis, and the manner in which B is tiled to construct the lattice. The real roots v = e ^{K} — 1 of P _{B}(q, v) either give the exact critical points for the lattice, or provide approximations that, in principle, can be made arbitrarily accurate by increasing the size of B in an appropriate way. In earlier work, P _{B}(q, v) was defined by a contractiondeletion identity, similar to that satisfied by the Tutte polynomial. Here, we give a probabilistic definition of P _{B}(q, v), which facilitates its computation, using the transfer matrix, on much larger B than was previously possible.We present results for the critical polynomial on the (4, 8 ^{2}), kagome, and (3, 12 ^{2}) lattices for bases of up to respectively 96, 162, and 243 edges, compared to the limit of 36 edges with contractiondeletion. We discuss in detail the role of the symmetriesmore »
 Authors:

^{[1]};
^{[2]}
 LPTENS, Ecole Normale Superieure, Paris (France); Univ. Pierre et Marie Curie, Paris (France)
 Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
 Publication Date:
 Report Number(s):
 LLNLJRNL610232
Journal ID: ISSN 17518121
 Grant/Contract Number:
 AC5207NA27344
 Type:
 Accepted Manuscript
 Journal Name:
 Journal of Physics. A, Mathematical and Theoretical (Online)
 Additional Journal Information:
 Journal Name: Journal of Physics. A, Mathematical and Theoretical (Online); Journal Volume: 46; Journal Issue: 7; Journal ID: ISSN 17518121
 Publisher:
 IOP Publishing
 Research Org:
 Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
 Sponsoring Org:
 USDOE
 Country of Publication:
 United States
 Language:
 English
 Subject:
 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; 97 MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE
 OSTI Identifier:
 1240068