Splitting-merging transitions in tensor-vectors systems in exact large-$N$ limits

TL;DR

This paper analyzes phase transitions in tensor-vector systems at large N, revealing cascades of first- and second-order transitions with implications for quantum gravity, data analysis, and spin glass models.

Contribution

It provides an exact large-N analysis of splitting-merging transitions in tensor-vector systems, identifying different phase transition types and symmetry breaking patterns.

Findings

Discovered cascades of first-order phase transitions for fixed tensors.

Identified first- and second-order transitions for random tensors.

Linked splitting dynamics to quantum gravity, data analysis, and spin glass models.

Abstract

Matrix models have phase transitions in which distributions of variables change topologically like the Gross-Witten-Wadia transition. In a recent study, similar splitting-merging behavior of distributions of dynamical variables was observed in a tensor-vectors system by numerical simulations. In this paper, we study the system exactly in some large-$N$ limits, in which the distributions are discrete sets of configurations rather than continuous. We find cascades of first-order phase transitions for fixed tensors, and first- and second-order phase transitions for random tensors, being characterized by breaking patterns of replica symmetries. The system is of interest across three different subjects at least: The splitting dynamics plays essential roles in emergence of classical spacetimes in a tensor model of quantum gravity; The splitting dynamics automatically detects the rank of a tensor in the tensor rank decomposition in data analysis; The system provides a variant of the spherical $p$-spin model for spin glasses with a new non-trivial parameter. We discuss some implications of the results from these perspectives. The results are compared with some numerical simulations to check the large-$N$ convergence and the assumptions made in the analysis.