Fermi Partition Functions of Friendly Walkers and Pair Connectedness of Directed Percolation.

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  • Inui Norio
    Department of Mechanical and Intelligent Engineering, Himeji Institute of Technology, 2167, Shosha, Himeji, Hyogo 671-2201
  • Katori Makoto
    Department of Physics, Faculty of Science and Engineering, Chuo University, Kasuga, Bunkyo-ku, Tokyo 112-8551

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Non-crossing random walkers with attractive interactions called friendly walkers (FWs) are studied. A restriction on trajectories, which is analogous to Pauli's exclusion principle, is imposed and the Fermi partition functions are defined. We prove a theorem that the pair connectedness of the bond directed percolation (DP) with bond concentration p is related to the Fermi grand partition function of FW if we set the temperature T=-1/(kB ln p) and the chemical potential μ=-i π/ln p, where kB is the Boltzmann constant and i=√-1. The pure imaginary chemical potential means that the DP transition can be regarded as a symmetry breaking of parity in the number of FWs. As a corollary of the theorem, a new method is proposed for calculating the series expansion of the pair connectedness and percolation probability of DP using the low-temperature expansion data of FW.

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