Heydenreich, Markus; Kolesnikov, Leonid
(2018):
The critical 1arm exponent for the ferromagnetic Ising model on the Bethe lattice.
In: Journal of Mathematical Physics, Vol. 59, No. 4, 43301

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Abstract
We consider the ferromagnetic nearestneighbor Ising model on regular trees (Bethe lattice), which is wellknown to undergo a phase transition in the absence of an external magnetic field. The behavior of the model at critical temperature can be described in terms of various critical exponents;one of them is the critical 1arm exponent rho which characterizes the rate of decay of the (root) magnetization as a function of the distance to the boundary. The crucial quantity we analyze in thiswork is the thermal expectation of the root spin on a finite subtree, where the expected value is taken with respect to a probability measure related to the corresponding finitevolume Hamiltonian with a fixed boundary condition. The spontaneous magnetization, which is the limit of this thermal expectation in the distance between the root and the boundary (i.e., in the height of the subtree), is known to vanish at criticality. We are interested in a quantitative analysis of the rate of this convergence in terms of the critical 1arm exponent rho. Therefore, we rigorously prove that <sigma(0)>(+)(n), the thermal expectation of the root spin at the critical temperature and in the presence of the positive boundary condition, decays as <sigma(0)>(+)(n) approximate to n(1/2) (in a rather sharp sense), where n is the height of the tree. This establishes the 1arm critical exponent for the Ising model on regular trees (rho = 1/2). Published by AIP Publishing.