Combinatorial Aspects of the Ising Model for Ferromagnetism. I. A Conjecture of Feynman on Paths and Graphs

摘要: An identity on paths in planar graphs conjectured by Feynman [H] is rigorously established. This permits a complete analysis of the combinatorial approach to two‐dimensional Ising model with nearest neighbor interaction and 0 external magnetic field previously heuristically discussed Kac Ward [KW] Potts [PW]. Relevant identities are established for next interactions field, positive three‐dimensional field. For case square net an odd number spin locations equal iπ/2, it shown that partition function identically zero both plane torus imbedding contrary result announced Lee Yang [LY; Eq. (48)], which turns out be correct only even locations.