2008/05/31 by V. A. Avetisov, V A Avetisov, A. Kh. Bikulov +3 · 1 citation
Mathematics · Physics and Astronomy · #Bounded function #Eigenvalues and eigenvectors #Gaussian #Partition function (quantum field theory) #Probability density function #Probability distribution #Random Matrices and Applications #Random matrix #Spectral Theory in Mathematical Physics #Stochastic matrix #Ultrametric space #advanced mathematical theories #cond-mat.dis-nn #cond-mat.stat-mech
paper · pdf · doi:10.1088/1751-8113/42/7/075001
7 pages, 2 figures (the paper is essentially reworked)
arxiv created 2008/12/09 · openalex publication_date 2009/01/21 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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.