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We study the statistical and dynamic properties of the systems characterizedby an ultrametric space of states and translationary non-invariant symmetrictransition matrices of the Parisi type subjected to "locally constant"randomization. Using the explicit expression for eigenvalues of such matrices,we compute the spectral density for the Gaussian distribution of matrixelements. We also compute the averaged "survival probability" (SP) having senseof the probability to find a system in the initial state by time $t$. Using thesimilarity between the averaged SP for locally constant randomized Parisimatrices and the partition function of directed polymers on disordered trees,we show that for times $t>t_{\rm cr}$ (where $t_{\rm cr}$ is some criticaltime) a "lacunary" structure of the ultrametric space occurs with theprobability $1-{\rm const}/t$. This means that the escape from some boundedareas of the ultrametric space of states is locked and the kinetics is confinedin these areas for infinitely long time.

Random Hierarchical Matrices: Spectral Properties and Relation to Polymers on Disordered Trees