Stochastic melonic kinetics with random initial conditions

TL;DR

This paper investigates the stochastic dynamics of tensor field theories, focusing on melonic sectors and phase transitions, using Langevin equations and large N limits, supported by numerical simulations.

Contribution

It introduces a stochastic approach to tensor field theories, analyzing melonic kinetics and phase transitions in the large N limit with numerical validation.

Findings

Identifies the transition temperature between equilibrium and non-equilibrium regimes.

Provides a general formula for the phase transition temperature.

Supports theoretical results with numerical simulations.

Abstract

The probability laws associated with random tensors or tensor field theories are traditionally equilibrium distributions. In this paper, we consider a stochastic point of view, and approach the quantization by a Langevin type equation. We especially address the low-temperature behavior of the phase ordering kinetics of a stochastic complex tensor field $T_{i_1\cdots i_d}(x,t)$ of size $N$ and rank $d$ in dimension $D$. The method we propose use the self averaging property of the tensorial invariants in the large $N$ limit. In this regime, the dynamics is governed by the melonic sector, whose behavior is studied in the quenched limit, where the contractions involving $d-1$ indices self-average around a diagonal matrix proportional to the identity. The following work especially focuses on the cyclic (i.e. non-branching) melonic sector, and we study the way that the system returns to the equilibrium regime regarding the temperature and the shape of the potential. In particular, we provide a general formula for the transition temperature between these regimes. The manuscript is accompanied by numerical simulations to support the theoretical analysis, and essentially aims to open towards this new field of investigation.