Construction of Non-Gaussian Self-Similar Random Fields with Hierarchical Structure

In the present work we construct non-Gaussian self-similar random fields with hierarchical structure. The construction is based on non-Gaussian solutions of the main nonlinear equation of the hierarchical models theory. The existence of such solutions was proved originally by Sinai and the author * and later by another method by Collet and Eckmann. Next we establish the uniqueness of a Gibbs state for the constructed self-similar field. Finally for a class of hierarchical models we prove the convergence of renormalization transformations of a random field at the critical point to the self-similar field. 1. Definitions Let reTL, r ^ 2 , and ξQ>ξι>ξ2>... be a decreasing sequence of partitions of a countable set V satisfying the following conditions: (i) ξ0 is the partition of V into separate points, (ii) any element of the partition ξn consists of r elements of the partition ξπ_1 ? n = l , 2 , . . . , (iii) for any two points iJeVa, number n exists such that ίj belong to the same element of the partition ξn. Such a sequence of partitions is called a hierarchical structure in the set V (see [1]). Let us denote n(ij) the least number n such that ij belong to the same element of the partition ξn. The quantity 0 ί f J ^ ) if defines a metrics in a set V with hierarchical structure. A map V-> V is called an isomorphism of hierarchical structures if it preserves the structure of the partitions. One can see easily that for a given r any two hierarchical structures are isomorphic. We shall consider two realizations of the 0010-3616/82/0084/0557/S04.40