# The hessian in spin foam models

## TL;DR

This paper analyzes the Hessian in EPRL spin foam models, demonstrating its nondegeneracy at geometric stationary points with spacelike faces, which advances understanding of the models' asymptotic behavior.

## Contribution

It provides the first proof of the nondegeneracy of the Hessian at key stationary points in EPRL models with all faces spacelike.

## Key findings

- Hessian is nondegenerate at geometric stationary points.
- Analysis applies to nondegenerate 4-simplices with spacelike faces.
- Fills a gap in the asymptotic analysis of EPRL spin foam models.

## Abstract

We fill one of the remaining gaps in the asymptotic analysis of the vertex amplitudes of the Engle-Pereira-Rovelli-Livine (EPRL) spin foam models: We show that the hessian is nondegenerate for the stationary points that corresponds to geometric nondegenerate $4$ simplices. Our analysis covers the case when all faces are spacelike.