Shapes of tree representations of spin-glass landscapes

TL;DR

This paper investigates the shape of barrier trees derived from spin-glass energy landscapes using phylogenetic tree statistics, revealing their ability to classify landscape types and their structural differences from random trees.

Contribution

It introduces the application of phylogenetic tree statistics to barrier trees of spin-glass models, demonstrating their effectiveness in landscape classification and structural analysis.

Findings

At least one statistic distinguishes landscape types

Barrier trees differ significantly from random trees

Subtree structure correlates with depth in the tree

Abstract

Much of the information about the multi-valley structure of disordered spin systems can be convened in a simple tree structure -- a barrier tree -- the leaves and internal nodes of which represent, respectively, the local minima and the lowest energy saddles connecting those minima. Here we apply several statistics used in the study of phylogenetic trees to barrier trees that result from the energy landscapes of p-spin models. These statistics give information about the shape of these barrier trees, in particular about balance and symmetry. We then ask if they can be used to classify different types of landscapes, compare them with results obtained from random trees, and investigate the structure of subtrees of the barrier trees. We conclude that at least one of the used statistics is capable of distinguishing different types of landscapes, that the barrier trees from p-spin energy landscapes are quite different from random trees, and that subtrees of barrier trees do not reflect the overall tree structure, but their structure is correlated with their ``depth'' in the tree.