The one-site distribution of Gibbs states on Bethe lattice are probability vectors of period⩽2 for a nonlinear transformation

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

$$a(r,s) = \exp (K[r,s] + (1/q)[N,r + s])$$

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.