A deletion-invariance property for random measures satisfying the Ghirlanda-Guerra identities

TL;DR

This paper proves a deletion-invariance property for discrete random measures satisfying the Ghirlanda-Guerra identities, showing that randomly deleting half of the points and renormalizing preserves the measure's distribution up to rotations.

Contribution

It establishes a novel invariance property of such measures under random deletion and renormalization, extending understanding of their structural symmetries.

Findings

Deletion-invariance property proven for measures satisfying Ghirlanda-Guerra identities

Random deletion and renormalization preserve the measure's distribution up to rotations

Enhances understanding of symmetries in spin glass models

Abstract

We show that if a discrete random measure on the unit ball of a separable Hilbert space satisfies the Ghirlanda-Guerra identities then by randomly deleting half of the points and renormalizing the weights of the remaining points we obtain the same random measure in distribution up to rotations.