Abstract
We prove that the one-site distribution of Gibbs states (for any finite spin setS) on the Bethe lattice is given by the points satisfying the equation π=T 2π, whereT=h·A·ϕ, withϕ(x)=x (q−1/q,h(x)=(x∥x∥ q )q,A=(a(r, s)∶r, s∈S), and
We also show that forA a symmetric, irreducible operator the nonlinear evolution on probability vectorsx(n+1)=Ax(n)p∥Ax(n)p∥1 withp>0 has limit pointsξ of period⩽2. We show thatA positive definite implies limit points are fixed points that satisfy the equationAξ p=λξ. The main tool is the construction of a Liapunov functional by means of convex analysis techniques.
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Goles, E., Martinez, S. The one-site distribution of Gibbs states on Bethe lattice are probability vectors of period⩽2 for a nonlinear transformation. J Stat Phys 52, 267–285 (1988). https://doi.org/10.1007/BF01016414
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DOI: https://doi.org/10.1007/BF01016414