Asymptotics in infinite monoidal categories
A Perron–Frobenius and random-walk viewpoint on tensor-power growth in monoidal categories with infinitely many indecomposables.
Data
- Title: Asymptotics in infinite monoidal categories
- Authors: Abel Lacabanne, Daniel Tubbenhauer and Pedro Vaz
- Status: Higher Structures 9(2): 168-197, 2025.
- Code / errata: Click
- arXiv: https://arxiv.org/abs/2404.09513
Abstract
We discuss formulas for the asymptotic growth rate of the number of summands in tensor powers in certain (finite or infinite) monoidal categories. Our focus is on monoidal categories with infinitely many indecomposable objects, with our main tools being generalized Perron–Frobenius theory alongside techniques from random walks.
What is the point?
Finite categories often behave like finite matrices. Infinite categories behave more like infinite graphs, so one has to understand whether random walks return or escape. This paper explains when the usual growth philosophy survives in the infinite setting.
fusion rules become walks.
recurrence controls the asymptotics.
Perron–Frobenius remains the guide.
More details
This paper extends our finite-category results to categories with infinitely many indecomposable objects. Which asymptotic results survive depends on properties of the fusion graph, formulated through generalized Perron–Frobenius theory and suitable recurrence assumptions.
A recurrent random walk returns to its starting point with probability \(1\). Positive recurrence additionally requires a finite expected return time. The simple symmetric random walk on \(\mathbb{Z}\) is null recurrent, not positively recurrent.
The plots below compare the numbers of paths of length \(200\) from the origin to each endpoint on \(\mathbb{Z}\) and on the half-line \(\mathbb{Z}_{\geq 0}\), illustrating how a boundary changes the distribution of endpoints.
