Block diagonalizing ultrametric matrices

TL;DR

This paper investigates the diagonalization of ultrametric matrices, which are relevant in disordered systems with multiple equilibrium phases, revealing a block diagonal structure that simplifies their analysis.

Contribution

It introduces a method to block diagonalize ultrametric matrices using residual symmetry, providing explicit forms for the blocks and their sizes, including the inverse matrices.

Findings

Ultrametric matrices can be brought to a block diagonal form by a common similarity transformation.

Most blocks are 1x1, effectively diagonalizing large sectors of the matrix.

Remaining blocks are of size (R+1) x (R+1), related to the number of replica symmetry breaking steps.

Abstract

The problem of diagonalizing a class of complicated matrices, to be called ultrametric matrices, is investigated. These matrices appear at various stages in the description of disordered systems with many equilibrium phases by the technique of replica symmetry breaking. The residual symmetry, remaining after the breaking of permutation symmetry between replicas, allows us to bring all ultrametric matrices to a block diagonal form by a common similarity transformation. A large number of these blocks are, in fact, of size 1 times 1, i.e. in a vast sector the transformation actually diagonalizes the matrix. In the other sectors we end up with blocks of size (R+1) times (R+1) where R is the number of replica symmetry breaking steps. These blocks cannot be further reduced without giving more information, in addition to ultrametric symmetry, about the matrix. Similar results for the inverse of a generic ultrametric matrix are also derived.