Complex Obtuse Random Walks and their Continuous-Time Limits

TL;DR

This paper introduces complex obtuse random variables, explores their algebraic structures, and demonstrates their convergence to normal martingales in continuous time, with implications for quantum statistical mechanics.

Contribution

It extends the concept of obtuse random variables to the complex case, characterizes them via a symmetric 3-tensor, and analyzes their continuous-time limits and associated operators.

Findings

Complex obtuse random variables are characterized by a symmetric 3-tensor.

They converge to normal martingales in complex space in the continuous-time limit.

The associated 3-tensor encodes the martingale's behavior, including diffusion and Poisson-like directions.

Abstract

We study a particular class of complex-valued random variables and their associated random walks: the complex obtuse random variables. They are the generalization to the complex case of the real-valued obtuse random variables which were introduced in \cite{A-E} in order to understand the structure of normal martingales in $\RR^n$.The extension to the complex case is mainly motivated by considerations from Quantum Statistical Mechanics, in particular for the seek of a characterization of those quantum baths acting as classical noises. The extension of obtuse random variables to the complex case is far from obvious and hides very interesting algebraical structures. We show that complex obtuse random variables are characterized by a 3-tensor which admits certain symmetries which we show to be the exact 3-tensor analogue of the normal character for 2-tensors (i.e. matrices), that is, a necessary and sufficient condition for being diagonalizable in some orthonormal basis. We discuss the passage to the continuous-time limit for these random walks and show that they converge in distribution to normal martingales in $\CC^N$. We show that the 3-tensor associated to these normal martingales encodes their behavior, in particular the diagonalization directions of the 3-tensor indicate the directions of the space where the martingale behaves like a diffusion and those where it behaves like a Poisson process. We finally prove the convergence, in the continuous-time limit, of the corresponding multiplication operators on the canonical Fock space, with an explicit expression in terms of the associated 3-tensor again.