Trees that can be grown in "too many" ways: A review of Bouch's construction

TL;DR

This paper reviews Bouch's hierarchical construction of lattice trees, highlighting their combinatorial growth and implications for operator dynamics in higher-dimensional quantum spin systems.

Contribution

It provides a detailed analysis of Bouch's tree construction method and discusses its significance for understanding operator growth in quantum many-body systems.

Findings

Trees can be grown in factorially many ways from the root.

The growth rate of these trees has implications for quantum operator spreading.

The construction links combinatorial tree growth to quantum system dynamics.

Abstract

We carefully review the hierarchical construction by Bouch [Bouch2015] of trees on the square lattice that can be grown from its root in $L!/C^L$ distinct ways, where $L$ denotes the number of bonds constituting the tree, and $C>1$ is a constant. (As discussed in Section IV.A of [ParkerCaoAvdoshkinScaffidiAltman2019] and Appendix A.3 of [ShiraishiTasaki2024], this result has an implication on the operator growth in quantum spin systems in two or higher dimensions.)