Stochastic processes with Z_N symmetry and complex Virasoro representations. The partition functions

TL;DR

This paper analytically confirms that a Z_N symmetric stochastic process, derived from a Hamiltonian related to the Temperley-Lieb algebra, exhibits a complex spectrum with oscillatory convergence to stationarity.

Contribution

It provides explicit analytical expressions for the partition functions of the system, validating previous numerical findings and revealing spectral degeneracies and oscillatory behaviors.

Findings

Partition functions confirmed analytically.

Spectral degeneracy for even N leads to two processes.

Complex spectrum causes oscillations in convergence.

Abstract

In a previous Letter (J. Phys. A v.47 (2014) 212003) we have presented numerical evidence that a Hamiltonian expressed in terms of the generators of the periodic Temperley-Lieb algebra has, in the finite-size scaling limit, a spectrum given by representations of the Virasoro algebra with complex highest weights. This Hamiltonian defines a stochastic process with a Z_N symmetry. We give here analytical expressions for the partition functions for this system which confirm the numerics. For N even, the Hamiltonian has a symmetry which makes the spectrum doubly degenerate leading to two independent stochastic processes. The existence of a complex spectrum leads to an oscillating approach to the stationary state. This phenomenon is illustrated by an example.