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

TL;DR

This paper investigates the spectral and dynamic properties of randomized Parisi matrices in ultrametric spaces, revealing a phase where system states become confined, akin to polymers on disordered trees, with implications for understanding complex systems.

Contribution

It provides explicit eigenvalue expressions and links the spectral properties of randomized Parisi matrices to the behavior of directed polymers on disordered trees, highlighting a confinement transition.

Findings

Spectral density computed for Gaussian matrix elements.

Identification of a critical time $t_{cr}$ for confinement transition.

Ultrametric space exhibits a lacunary structure after $t_{cr}$.

Abstract

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.