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

V. A. Avetisov 1, A. Kh. Bikulov 2, S. K. Nechaev 3

Journal of Physics A Mathematical and Theoretical 42 (2009) 075001

We study the statistical and dynamic properties of the systems characterized by an ultrametric space of states and translationary non-invariant symmetric transition 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 matrix elements. We also compute the averaged 'survival probability' (SP) having sense of the probability to find a system in the initial state by time $t$. Using the similarity between the averaged SP for locally constant randomized Parisi matrices 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 critical time) a 'lacunary' structure of the ultrametric space occurs with the probability $1-{\rm const}/t$. This means that the escape from some bounded areas of the ultrametric space of states is locked and the kinetics is confined in these areas for infinitely long time.

• 1. The Semenov Institute of Chemical Physics,
Russian Academy of Sciences
• 2. The Semenov Institute of Chemical Physics,
Russian Academy of Science
• 3. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
CNRS : UMR8626 – Université Paris XI - Paris Sud