# Uniqueness and Non-Uniqueness for Spin-Glass Ground States on Trees

## TL;DR

This paper investigates the conditions under which ground states of spin glasses on trees are unique or non-unique, linking this to flow and recurrence properties of the underlying graph.

## Contribution

It establishes a novel equivalence between ground state uniqueness, maximal flow, and recurrence of random walks on trees for spin glasses.

## Key findings

- Uniqueness of ground states corresponds to zero maximal flow.
- Non-uniqueness occurs when the maximal flow is positive.
- Ground state uniqueness is linked to recurrence of the simple random walk.

## Abstract

We consider a Spin Glass at temperature $T = 0$ where the underlying graph is a locally finite tree. We prove for a wide range of coupling distributions that uniqueness of ground states is equivalent to the maximal flow from any vertex to $\infty$ (where each edge $e$ has capacity $|J_{e}|$) being equal to zero which is equivalent to recurrence of the simple random walk on the tree.