ISSN:
1572-9613
Keywords:
Gibbs states
;
Bethe lattice
;
spin vector
;
dynamical systems
;
automata networks
;
subdifferential
;
cyclically monotone function
;
convex function
;
Liapunov functional
Source:
Springer Online Journal Archives 1860-2000
Topics:
Physics
Notes:
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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01016414
