Large deviations for hierarchically dependent random variables (Q788407)

The author considers a class of interacting fields in statistical physics, Dyson's hierarchical models in the unbounded spin case. In this model, the \(n^{th}\) ''block'' is defined by \[ P\{\xi_ 1^{(n)}\in dt_ 1,...,\xi_{2^ n}^{(n)}\in dt_ 2n\}=\Xi^{-1}\exp \{-\beta H_ n(t_ 1,...,t_{2n})\}\prod^{2^ n}_{i=1}\mu_ 0(dt_ i), \] where the Hamiltonian \(H_ n\) shows the hierarchical block interaction between spins. Using the renormalization group equation he first estimates the density function \(p_ n\) of the \(n^{th}\) arithmetic mean \(\eta_ n=2^{-n}\sum^{2^ n}_{k=1}\xi_ k^{(n)}\) under certain conditions on regularity and symmetry of the initial probability \(\mu_ 0\). From there he derives the main results of the paper: A local limit theorem and asymptotic expansion of large deviations for the distribution density \(p_ n\).