Generic Ising trees (Q2890743)

The authors study the Ising model with an external magnetic field on certain infinite random trees. The authors prove that the generic Ising tree exhibits no single site spontaneous magnetization at the root \(r\) or at any other spine vertex. The authors establish the existence of the measure \(\mu^{(\beta,h)}\) on the set of infinite trees for given values of the coupling constants \(\beta,h\). Also, they determine the asymptotic behavior of the partition functions of ensembles of spin systems on finite trees of large size. Considering the dependence of the partition function on the size of trees they study the Ising model on finite but large trees. The limiting measure is studied. The existence of the weak limits is proved under the genericity condition. The authors determine the values of the Hausdorff and spectral dimensions of the ensemble of trees obtained from a vertex \(v\) by integrating over the spin degrees of freedom. The authors show that the annealed Hausdorff dimension of a generic Ising tree can be evaluated and equals that of generic random trees as introduced in [\textit{B. Durhuus, T. Jonsson} and \textit{J. F. Wheater}, J. Stat. Phys. 128, No. 5, 1237--1260 (2007; Zbl 1136.82006)]. The authors calculate the values of the Hausdorff and spectral dimensions of the underlying trees for some parameters. Under some assumptions, the annealed spectral dimension of given trees is calculated. The magnetization properties of generic Ising trees in some detail are discussed. In view of the fact that the trees have a single spine, the authors distinguish between the magnetization on the spine and the bulk magnetization. Also, they establish the absence of magnetization. Under some assumptions, the authors prove that the mean magnetization on the spine vanishes. They define the connected two-point function. Applying the given techniques, they find new two-point functions. Finally, some concluding remarks on possible future developments are collected.