Space-time percolation

TL;DR

This paper explores space-time percolation models, including directed and undirected variants, and their applications to quantum Ising models, providing new insights into phase transitions, entanglement bounds, and connections to random graphs.

Contribution

It introduces a continuum space-time percolation framework for quantum Ising models and related processes, extending classical models and offering new analytical tools.

Findings

Analysis of contact model at criticality using percolation techniques

Construction of a random-cluster model on space-time lattice

Bound on entanglement entropy in quantum Ising model

Abstract

The contact model for the spread of disease may be viewed as a directed percolation model on $\ZZ \times \RR$ in which the continuum axis is oriented in the direction of increasing time. Techniques from percolation have enabled a fairly complete analysis of the contact model at and near its critical point. The corresponding process when the time-axis is unoriented is an undirected percolation model to which now standard techniques may be applied. One may construct in similar vein a random-cluster model on $\ZZ \times \RR$, with associated continuum Ising and Potts models. These models are of independent interest, in addition to providing a path-integral representation of the quantum Ising model with transverse field. This representation may be used to obtain a bound on the entanglement of a finite set of spins in the quantum Ising model on $\ZZ$, where this entanglement is measured via the entropy of the reduced density matrix. The mean-field version of the quantum Ising model gives rise to a random-cluster model on $K_n \times \RR$, thereby extending the Erdos-Renyi random graph on the complete graph $K_n$.